T = temperature T = average temperature X = matrix containing partial derivatives Y n = vector containing n rate observations g n l = predicted value of model i at Fn Greek Letters q = effectiveness factor u2 = error variance = vector of unknown parameters in model i f = vector containing independent variables specifying experimental conditions Literature Cited Barkiey, L. W., Corrigan. T. E., Wainwright, H. W., Sands, A. E.. lnd. Eng. Chem., 4 4 ( 5 ) , 1066 (1952). Blakemore, J. W.. Hoerl, A. E., Chem. Eng. Prog. Symp. Ser., 59(42), 14 (1963). Bohlbro, H., Acta. Chem. Scand., 15, 502 (1961). Bohlbro, H.. Acta. Chem. Scand. 16, 431 (1962). Bohlbro. H.. Acta. Chem. Scand., 17, 1001 (1963). Bohibro, H., J. Catal., 3, 207 (1964). Bohibro, H.. "An Investigation on the Kinetics of the Conversion of Carbon Monoxide with Water Vapour Over iron Oxide Based Catalysts," Gjellerup. 1966. Boreskov. G. K., Yur'eva, T. M., Sergeeva. A . S . . Kinet. Katal., 11(6). 1476 (1970). Box, G. E. P., Henson. T. L., M. B. R. Technical Report No. 51, University of Wisconsin, 1969. Box, G. E. P.. Hill, W. J . , Technometrics, 9 ( 1 ) , 57 (1967).
Box, G. E. P.. Kanemasu. H., Technical Reports No. 320, 321, 322, 323, Department of Statistics, University of Wisconsin, 1972-1973. Box, G. E. P.. Lucas. H . , Biometrika, 46, 77 (1959). Froment, G., Mezaki, R., Chem. Eng. Sci., 25, 293 (1970). Giona, R., Passino, R., Toselli, L'lngegnere, 33, 631 (1959); Chem. Abstr., 55, 1157e(1961). Hosten, L., PhD. Dissertation, Rijksuniversiteit te Gent, Gent, Belgium, 1967. Hulburt, H. M.. Vasan.C. D . Sc.,A.l.Ch.E. J.. 7, 143 (1961). Kaneko, Y . . Oki, S., J. Res. Inst. Catal. Hokkaido Univ., 1 3 ( 1 ) , 55 (1965a). Kaneko, Y., Oki, S., J. Res. Inst. Catal. Hokkaido Univ.. 1 3 ( 3 ) 169 (1965b). Kodama, S., Fukui. K.. Tame, T.. Kinoshita, M.. Shokubai, 8, 50 (1952): Chem. Abstr., 47, 11920 (1953). Kul'kova, N. V.. Temkin, M . I., Zh. Fiz. Khim., 23, 695 (1949). Mezaki, R., Kittrell. J. R., lnd. Eng. Chem.. 5 9 ( 5 ) , 63 (1967). Oki, S..Happel. J., Hnatow. M. A.. Kaneko, Y.. Proc. Int. Congr. Catal., 5th, Aua (1972). Oki, S..Mezaki, R . , J. Phys. Chem., 77, 447 (1973a). Oki, S.,Mezaki, R., J. Phys. Chem., 77, 1601 (1973b). Paratella, A.. Chim. lnd., 47, 38 (1965); Chem. Abstr., 63, 400f (1965). Satterfield, C. N., "Mass Transfer in Heterogeneous Catalysis." M.I.T. Press, pp 21-27, Cambridge, Mass., 1970. Shchibrya, G. G., Morozov. N. M., Temkin, M. I., Kinet. Katal, 6 ( 6 ) , 1057 (1965). Weisz. P. B.. Prater, C. D., Advan. Catal.. 6 167 (1954). Yur'eva, T. M. Boreskov, G. K., Graver, V. S., Kinet. Katal., 1 0 ( 4 ) , 862 (1969).
Received for review November 16, 1973 Accepted June 14,1974
Synthesis of Cascade Refrigeration and Liquefaction Systems Francisco J. Barnes' and C. Judson King* Department of Chemical Engineering. University of California. Berkeley, California 94720
Systematic synthesis procedures are employed to seek minimum-cost process configurations for cascade refrigeration and gas liquefaction systems. Through a preliminary process analysis based upon graph decomposition principles it is shown that the optimal or near-optimal configuration is defined by the answers to a limited number of identifiable choices, involving both configuration and design variables. The minimum-cost configuration is located through the application of dynamic programming to the network formed by these alternatives. The procedure is repeated iteratively as optimal values are found for design variables and the number of refrigerant levels. Certain heuristics are used or implied in this overall iterative procedure, but they do not appear to limit the method significantly. Several practical examples of synthesis of cascade refrigeration and gas liquefaction systems are considered. The results are interpreted to infer factors that appear to be dominant for refrigeration system design.
Introduction
Process design may be described as the conception or synthesis of a given process configuration, followed by the evaluation or analysis of its capabilities, equipment sizes, and cost requirements. Most often, the design process is evolutionary, following a succession of alternative steps of synthesis and analysis, wherein the results of the last analysis provide additional information to improve the next synthesis (King, e t al., 1972). Whereas process analysis has received a great deal of attention and has achieved a high level of sophistication, only recently have substantial efforts been initiated to structure the logic of synthesis of chemical processes (Hendry, et al., 1973). The bulk of this research has fol-
' Department of Chemical Engineering. National Autonomous Universityof Mexico, Mexico 20, D.F.
lowed the availability of large computers making extensive use of their large memory capacity and high speed. Morphological analysis and other more qualitative approaches have also been utilized for structuring process understanding and synthesis for more broadly defined problems (King, 1973). Systematic process synthesis for processes involving sequentially connected elements may be defined as the orderly generation and evaluation of all possible alternative process configurations, followed by the application of efficient methods to discard as soon as possible the configurations that are less attractive than those remaining for further evaluation. These methods can be based upon either rigorous tests or mathematical algorithms that necessarily lead to the optimal solution, or upon heuristic methods. Heuristics are rules-of-thumb which cannot be shown to lead necessarily to the optimal solution, and the efficiency which they give to the search method must be balanced I n d . Eng. Chem.,
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against the presumed likelihood of their leading to an optimal or near-optimal solution. Refrigeration systems are attractive for studies of systematic synthesis, not only because of their economical importance in numerous key processes but also because they incorporate most of the energy transfer operations in widespread use in the chemical industry. The results obtained in this work may prove of value for the synthesis of even more complex processes where thermodynamic efficiency is important. The systems selected for study are mechanical refrigeration and gas liquefaction, since they have the advantage of involving no reaction or separation steps, keeping a relative simplicity. In addition, design decisions are based primarily upon thermodynamics, which has few and powerful rules. These systems have a simple cycle representation on a pressure-enthalpy diagram. This offers an opportunity to explore the potential for synthesis methods based on simple concepts of network theory, which have proved to be very effective when applied to the solution of synthesis problems in other areas (Kaufman, 1967; Potts and Oliver, 1972). Menzies and Johnson (1972) considered the problem of energy transfer within a process requiring refrigeration, and to a lesser extent transfer between the process and the refrigeration cycle, while keeping the design of the refrigeration cycle fixed. The present work is addressed to the problem of energy transfer within the refrigeration system and to a lesser extent between the process and the refrigeration system, while maintaining the design of the process fixed. Both methods are therefore complementary and may be used sequentially in an evolutionary way.
ValYe
B
r -
Process Stream
Q
Compressor
Vaporizer
Figure 1. Mechanical refrigeration-simplest
cycle,
Mechanical Refrigeration In the simplest refrigeration cycle a refrigerant vapor is compressed to a pressure high enough to be condensed with water or air, and the liquid formed is expanded through a valve, undergoing a partial vaporization that lowers the temperature. The remaining liquid is vaporized in a heat exchanger to provide the desired refrigeration, and the vapor is then returned to the compressor, as shown in Figure 1. In this process the lowest temperature obtained in the cycle is limited by the lowest acceptable compressor intake pressure, often selected to be slightly above atmospheric pressure. When refrigeration at lower temperatures is required, a cascade system is necessary, consisting of two or more independent cycles with the coldest refrigerant condensing against a warmer refrigerant instead of cooling water. Cycles involving compression of the vapor above the critical point will not be considered. Several modifications are possible to improve the performance of this simple cycle. Those considered in this work are shown in Figure 2, together with their representation on a pressure-enthalpy diagram. Economizers. The condensed refrigerant may be expanded to an intermediate pressure, where the vapor formed during expansion is separated from the remaining liquid before the liquid is further expanded. A lower compression ratio can be used to bring that vapor to the condensation pressure, thereby reducing compression work. Sideloads. Unless all the refrigeration demand is required at the lowest temperature, some of the demand may be satisfied with an intermediate refrigeration level, obtained by expanding some of the condensed refrigerant to an intermediate pressure, and evaporating it in a second vaporizer, with the resultant vapor admitted to an intermediate point in the compressor. This modification can 422
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H -
Figure 2. Possible process operations in a refrigeration cycle.
be combined with the use of economizers so that the liquid required for the intermediate refrigeration is withdrawn from the economizer, as shown in Figure 2. Aftercoolers. Particularly in the case of cascade refrigeration systems, it is often possible to pre-cool the final compressed vapor before the condenser with a warmer and less expensive refrigerant than that used in the condenser. Intercoolers. When the temperature of a partially compressed vapor is reduced by cooling it with water or other available refrigerant before further compression, the power consumption and the aftercooling duty are reduced. Presaturators. Another possible modification to save compressor work is presaturation of the partially compressed refrigerant vapor after an intermediate compression step by evaporating liquid from the corresponding economizer. This again lowers the temperature and specific volume of the vapor entering the next step of compression. Vapor-Liquid Heat Exchangers. The last modification considered in this work is the possibility of heat exchange between vapor and liquid streams in the refrigeration system. In these heat exchangers the vapor before compression is superheated while subcooling the liquid before expansion, so that the refrigeration obtained from the vapor is recovered when the liquid is evaporated after expan-
Table I. Materials of Construction Material of construction
Relative cost f o r compressors
Relative cost f o r heat exchangers
Lower temperature limit
Carbon s t e e l Killed carbon s t e e l 3% Nickel s t e e l 9% Nickel s t e e l
1.00 1.29 1.43 1.65
1.00 1.43 2.12 3.00
245 K 225 K 170 K 115 K
sion. This change will increase the amount of refrigeration per unit of refrigerant flow in the cycle, while increasing the work of compression and cooling load per unit flow. A second potential advantage is obtained in those cycles where expensive materials of construction are required in the compressor if the vapor is admitted at its saturation temperature; in this case heating the vapor to a high enough temperature may allow the use of a cheaper material of construction for the compressor. It is also possible to use vapor in a cold cycle to cool liquid in a warmer cycle of a cascade system.
Table 11. Equipment Cost Factors
Gas Liquefaction Gas liquefaction may be considered as a modification of a refrigeration cycle, incorporating a feed inlet at the proper location and a final vapor-liquid separator instead of a vaporizer. The liquid product is withdrawn from the cycle in this separator, while the vapor formed during expansion is returned to the compressor.
where x and y are different constants for each type of equipment and S is a size parameter. The values used in this work for heat exchangers, compressors, and steam turbines are given in Table I1 (AIChE Student Contest Problem, 1959; Bressler, 1966; Swearingen, 1970). The cost of other items of equipment, such as separation drums, presaturators and expansion valves, was neglected. The fixed capital investment is taken to be a function of P for each item through a cost factor which allows for instrumentation, piping, insulation, installation, etc. (Peters and Timmerhaus, 1968), corrected by use of actual cost data for a natural-gas liquefaction plant (Bourquet, 1970). The resulting equation (Barn&, 1973) for the annual cost of capital, assuming an interest rate of 25% on the fixed capital investment and a depreciation fund to replace the equipment in 10 years, is
Problem Specifications Equipment. The equipment items considered in the present work are: (1) condensers, vaporizers, inter- and after-coolers and vapor-liquid heat exchangers; (2) isenthalpic expansion valves; (3) compressors; (4) economizers and final vapor-liquid separators; (5) presaturators; and (6) mixers (stream junctions). The system is simplified by assuming that all heat exchangers operate counter-currently with a single pass and a negligible pressure drop. The driving force is given by the logarithmic mean of the extreme temperature differences. In the case of coolers and condensers, these temperature differences are based on a constant refrigerant temperature, neglecting for this purpose the temperature rise of cooling water. Desuperheating, condensation, and subcooling are assumed to take place in separate exchangers. Multiple-stream heat exchangers are not considered. The heat transfer coefficient for condensers and vaporizers is assumed to be 0.51 kW/m2 K (90 Btu/hr f t 2 "F) and that for coolers and vapor-liquid heat exchangers 0.17 kW/m2 K (30 Btu/hr ft2 "F) (Ludwig, 1965). The available heat sink is cooling water at 300 K . Each cycle is assumed to operate with a single-casing multi-stage centrifugal compressor with an isentropic efficiency of 8070,driven by a noncondensing stream turbine with throttle steam a t 600 psia and exhaust a t 4 in. Hg gauge, and an efficiency of 75% (Steen-Johnsen, 1967). Materials of Construction. The selection of materials for each equipment item is controlled by the lowest temperature to which the equipment is exposed. The materials considered, their relative costs, and their lower allowable temperature limits are given in Table I (AIChE Student Contest Problem, 1959; Perry, 1963). Process Cost. The purchased cost of carbon steel equipment, P, in dollars is given by the exponential expression
Equipment
S
Units
x
Y
Heat exchanger Compressor Steam turbine
Area Work Work
m2 kW kW
600 2750 1050
0.6 0.5 0.6
(11
P = xsy
A = [0.95
+ 0.15(N
-
1)]C[l+ i
0 . 6 5 ( f i - 1 ) ] P , (2) where N is the number of levels of refrigerant in the cycle and f L is the relative cost of the material of construction for each item of equipment, i, taken from Table I. More levels increase the cost factor because of the increased complexity of piping, and more expensive materials decrease the cost factor since items such as instrumentation and insulation are not dependent upon the material used for the major equipment items. The operating costs used are $16/kW yr for cooling water (Peters and Timmerhaus, 1968) and $90/kW yr for compression energy (Steen-Johnsen, 1967, Stettenberg, 1972). The objective function, to be minimized, is the sum of the annual cost of capital and operating costs. Thermodynamic Properties. Thermodynamic properties are obtained following a two-step operation whereby the ideal gas properties are first calculated at the desired temperature using a polynomial expression for heat capacity (BarnBs, 1973), and then the deviations from ideal gas behavior are obtained as a function of pressure using a modified form of the Redlich-Kwong equation of state ( B a r d s , 1973), similar to that proposed by Soave (1972). In those cases where either pressure or temperature is not known, an iterative solution is required. This procedure is followed for both vapor- and liquid-phase conditions. Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
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General Synthesis Strategy The number of possible process configurations for a multilevel or cascade system is quite large, in view of the number of potential heat-exchange relationships, and is infinite if there may be any number of refrigerant levels. Therefore, as an initial step the number of alternatives to be searched is restricted through the use of design rules obtained from a preliminary process analysis based upon decomposition of cycles. These rules allow immediate rejection of alternatives not meeting required criteria for optimality. Some of these design rules limit configurational alternatives; others place limits on the values which may be taken on by the design variables. This preliminary analysis initially derives rules for the limiting case where operating costs dominate and equipment costs are negligible, and then considers the effects of departures from this idealized case. The remaining alternatives which satisfy the criteria from the design rules can be represented by a graph, or network, of allowable nodes, or thermodynamic states, and interconnections. An efficient method is developed to generate this network of remaining alternatives and to locate the optimal process configuration by the application of dynamic programming to the network. The optimal configuration is defined as that giving the minimum annual cost for the prevailing set of design parameters (temperature approaches in exchangers, pressures of intermediate levels of refrigeration, etc.). An iterative or evolutionary design procedure is used, involving repetitions of the network analysis and configurational optimization. This fulfills the needs for (1) initializing and optimizing the design parameters and (2) allowing for the nonlinear dependence of equipment costs upon capacity-values of y other than unity. An outer loop is used to optimize the number of levels used for each refrigerant. Heuristics are used or implied at certain points in this iterative procedure but do not appear to limit the method significantly. The design strategy was implemented totally on the computer for two reasons. First, it would be prohibitively time-consuming to make the necessary computations by hand, and second, the computer implementation forces a very detailed description of all the steps in the logic, making it easier to detect missing elements or faulty decisions. The main portions of the synthesis procedure are described in the next three sections. Limitation of Alternatives Cycle Decomposition. Some simple concepts of graph theory can be used t o help with the analysis of refrigeration and liquefaction cycles. A simple cycle will be defined as a closed cycle where the same amount of refrigerant is flowing at every point, Le., a cycle without intermediate levels, since no splitting or mixing of streams is allowed. Figure 1 is a representation of a simple cycle. A complex cycle will be defined as a closed cycle with splitting or mixing of streams, such as that shown in Figure 2. Conceptually a complex cycle may be considered as a superposition of simple cycles operating in a reversible cascade, the warmer cycles providing the refrigeration required to cool and condense the compressed vapor from the next cold cycle. Thus, the twolevel cycle shown in Figure 2 may be represented by a cascade system of two independent cycles with an infinitesize heat exchanger providing the cooling and condensation of the colder cycle while vaporizing the refrigerant in the warmer one. Presaturators and intercoolers are there424
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fore conceptually equivalent to aftercoolers, and economizers to condensers. A liquefaction cycle is an open cycle with a vapor feed and a liquid product. The vapor feed is assumed to be saturated at condensation pressure. A liquefaction cycle can be decomposed into a closed refrigeration cycle and a process stream which passes through a condenser cooled with an external refrigerant, and then through one or more expansion and recondensation steps. The refrigeration required for the condensation of the process stream is reversibly provided by the closed refrigeration cycle. , Design Rules-Simple Ideal Cycles. An ideal cycle is defined as one for which equipment costs are assumed to be negligible. The following design rules may be derived for simple ideal cycles. a. Condenser. For fixed upper and lower pressure levels and compressor inlet temperature, the work of compression W and the aftercooling heat duty QA per unit flow of refrigerant are fixed. Increasing the condenser duty QC will increase the refrigeration available in the vaporizer Qv by the same amount, since according to the first law of thermodynamics QA + Qc = Qv + W Since the refrigerant used in the condenser is less expensive than that used in the vaporizer, the condenser duty should remove as much heat per unit flow of refrigerant as possible. Rule 1. The refrigerant vapor should be totally condensed before expansion. Subcooling the liquid before expansion is also effective for increasing the heat removed in the condenser, but the next design rule will prevent subcooling with external refrigeration. b. Compressor. The liquid-phase enthalpy is nearly independent of pressure, except very near the critical point. Therefore compressing the vapor beyond the minimum pressure at which condensation is possible increases both the work of compression and the heat that must be removed in the aftercooler, without increasing the amount of refrigeration obtained from the cycle. Rule 2. The vapor should not be compressed beyond that pressure at which condensation is possible with the available coolant. c . Vapor-Liquid Heat Exchangers. Another way of increasing the amount of refrigeration obtained from a cycle is to use a heat exchanger to subcool the liquid before expansion by superheating the low-pressure vapor. If the assumptions are made that there is only one heat sink available for cooling and condensing the compressed vapor, and that the vapor is an ideal gas with a constant heat capacity, then it is possible to demonstrate (see Appendix) that the vapor-liquid heat exchanger will be effective in reducing the operating cost of the refrigeration cycle only if the following criterion is met
? > 1 where Cp is the specific heat of the vapor, T, is the condensation temperature, and X is the latent heat at T,. The exit vapor temperature from the exchanger should be made as high as is allowed by the second law of thermodynamics, if the criterion is met. In Figure 3 the value of (C,T,/X) - 1 for several refrigerants is shown as a function of condensation temperature, with Cp evaluated at T,. The term may be positive or negative depending upon the proximity of the condensation temperature to the critical temperature. For common operating conditions the term will be negative for
P
CpT/X- I
F: Final Node 1: Initial Node H = High Pressure Node Pressure Node
L :Low
H
-08
1 -300
- 200
- 100
I 100
0
Figure 4. Network of reachable nodes. Intermediate-level nodes are a t the same time H and L nodes. F, H, and H at lowest pressure nodes refer to n + 2 simple cycle.
T (OF)
Figure 3. (C,Tc/X) - 1 us. temperature for various refrigerants.
light hydrocarbons and ammonia and positive for the heavier hydrocarbons. A more general expression has been obtained (Barn&, 1973) for use when the vapor does not behave ideally or when it is used to subcool the liquid condensed in warmer cycles. However, in the general case there will be more than one heat sink available for cooling the vapor after compression, and it will be necessary to evaluate all possible alternatives defined by the next design rule. Rule $3. If a vapor is superheated before compression, this must be done by cooling the liquid streams in order of increasing temperature, and in each operation the exit vapor temperature should be that of the inlet liquid stream. This rule defines the sequence of heat-exchange operations and the vapor exit conditions, but does not define how many of these operations must take place. d. Aftercoolers and Intercoolers. In the general case where more than one heat sink may be used to cool the compressed vapor before saturation, the following design rule must be followed to reduce the operating cost of an ideal cycle. Rule 4 . If a particular refrigerant is used to cool a compressed or partly compressed vapor toward saturation, all potential refrigerants of warmer temperatures must be used in previous exchangers to cool that vapor. The vapor should be completely cooled in each exchanger to the temperature of the refrigerant used. e . Initial Intercooler. A corollary of Rule 4 is the following rule. Rule 5 . The vapor effluent from a compressor must be cooled a t least to the temperature of the next colder refrigerant. Design Rules-Complex Ideal Cycles. a. Mixers. When a refrigeration cycle has more than one level of refrigeration, the vapor produced a t each intermediate level must be mixed with the partially compressed vapor from colder levels before further compression. Since mixing two vapor streams a t different temperatures increases the entropy of the system, reducing its thermodynamic efficiency, an extension of Rules 3 , 4 , and 5 is Rule 6. Rule 6. Both the vapor stream produced a t an intermediate level and the partially compressed vapor stream from colder levels must be brought to the temperature of one of the available refrigerants. This will be the same temperature for both streams.
The temperature will be the same because of the lack of mixing inefficiency and because, following generalizations of the criterion given by eq 4 for ideal cycles, it will be advantageous to intercool compressed vapor to a given temperature if it is not advantageous to preheat vapor above that temperature before introduction to the next stage of compression. For this same reason, another corollary is that use of a presaturator with the vapor compressed to an intermediate level will increase the efficiency of a cycle if and only if it is not economical to superheat the vapor produced in that level. b. Economizers. Finally, Rule 1, requiring that the refrigerant be totally condensed before expansion, must be satisfied by all the simple cycles of the cascade into which a complex cycle may be decomposed. Rule 7. Economizers must be provided a t all intermediate pressures to separate the vapor produced by the partial expansion of higher-pressure liquid before the remaining liquid is further expanded. Representation of Process Alternatives-Ideal Cycles. A network of nodes is defined as a matrix, such that each node represents the thermodynamic properties of the vapor refrigerant a t the specified pressure of one of the refrigeration levels or the condensation pressure, and at the saturation temperature of one of these available refrigeration levels, as shown in Figure 4. If two or more refrigerant fluids are used in a cascade system, then saturation temperatures corresponding to the specified pressure levels of other refrigerant fluids should he incorporated as nodes for a given pressure level of any one fluid. The system is divided into constituent simple cycles, and each simple cycle is divided into low-pressure nodes, denoted by L in Figure 4 and corresponding to the vaporization pressure of that cycle, and high-pres$ure nodes, denoted by H in Figure 4 and corresponding to the condensation pressure of that cycle. Nodes a t intermediate pressure levels are a t the same time low-pressure nodes for one simple cycle and high-pressure nodes for another. The initial node, denoted by I in Figure 4, for each simple cycle is defined as the node corresponding to the saturation condition of the low-pressure vapor. The final node, denoted by F in Figure 4. is defined as the node representing the inlet conditions to the compressor of the next warmest simple cycle. For the warmest simple cycle, the final node is the saturated vapor entering the condenser. Each node thereby represents the properties of the refrigerant vapor a t the exit of one or more of the possible process operations. The effects of the limitation of alterInd. Eng. C h e m . , P r o c e s s D e s . Develop., Vol. 13, No. 4 , 1974
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natives in the preliminary process analysis have been to restrict these possible exit conditions to certain specific points rather than a continuum of conditions and to restrict the possible interconnections between nodes to a limited number of unidirectional connectors. From the design rules for ideal cycles, there are allowable directional connectors from each low-pressure node to a high-pressure node through the combination of a compression and an initial aftercooler. There are also allowable directional connectors from each low-pressure node to the next warmer low-pressure node through a vapor superheater, except for the node at highest temperature. Allowable directional connectors extend from each highpressure node to the next cooler high-pressure node through aftercoolers, intercoolers, and presaturators. If the simple cycles are numbered starting with the warmest one, the number of possible configurations for simple cycle n is n 1, and the total number of process configurations for a system with Nsimple cycles is ( N l)! There is a corresponding matrix of points for conditions attainable by liquid streams, but the liquid conditions are uniquely fixed by the sequence of vapor points included in the process. Design Rules-Real Cycles. Several modifications to the design rules for ideal cycles occur when they are extended to real cycles with significant equipment costs. a. Temperature Approach in Heat Exchangers. The first modification that must be introduced is to allow for a finite temperature approach in the operation of any condenser, cooler, or heat exchanger in the cycle. This is done by specifying in advance values for the minimum temperature approach of the heat-exchange operations leading to each node, so that the basic network of attainable nodes for ideal cycles is preserved. The actual exit conditions are then easily calculated as deviations from the node conditions. The values for the minimum temperature approaches are updated in the iterative portion of the method. b. Vapor Mixing. When equipment cost is a significant contribution to the process cost it is no longer necessary that Rule 6 be valid, not only because of the presence of finite temperature approaches in the exchangers but also because it may be economical to eliminate some heat exchange operations to reduce the capital cost at the expense of an increase in operating costs. The problem is handled by Rule 6a, which introduces a new possible operation of mixing of the vapor a t any of the high-pressure nodes in the next colder simple cycle with vapor corresponding to the optimal ideal-cycle compressor inlet node for the warmer simple cycle. The mixing operation represents a possible change in the compressor inlet conditions for the warmer cycle because of the finite difference in temperature between the two mixed vapor streams. c. Coolers and Presaturators. The policy of cooling a vapor using the cheapest refrigerant available (Rule 4) is no longer necessarily optimal. Using a more expensive refrigerant has the advantage of requiring a smaller number of exchangers with a reduced total area. It is necessary to modify Rule 4 to Rule 4a which adds new possible connections between each high-pressure vapor node and all of the colder nodes, representing these operations. d. Initial Cooler. For the same reason it is no longer true that the optimal policy requires cooling a vapor after compression to the temperature of the next coldest node, especially if the exit conditions are very close to those of that node, since this requires using a heat exchanger with a small temperature approach. It is then necessary to con-
+
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sider a second alternative consisting of treating the compressed vapor with the same policy as that used for the vapor at the next colder node (Rule 5a). e. Vapor-Liquid Heat Exchangers. Similarly, it is now necessary to consider the possibility of cooling all warmer refrigerant liquids in violation of the sequence of cooling portion of Rule 3. This requires the addition of new connections leading from each low-pressure vapor node to all warmer low-pressure nodes (Rule 3a). f. Compressors. If the different stages of compression in a complex cycle are carried out in a single compressor housing with inlet and outlet nozzles for intermediate levels, a new problem arises when different materials of construction may be selected for the compressor. Since a complex cycle will be synthesized proceeding from the high-pressure level to the low-pressure level, as shown in the next section, the inlet temperature to the compressor is not known until the whole cycle has been synthesized, yet this temperature governs the materials requirement. The solution to the problem is to allow possibilities of different materials of construction for the compressor. For a given material those compression operations with inlet temperatures lower than the minimum temperature permitted for that material are not allowed (Rule 5b). Representation of Process Alternatives-Real Cycles. The design-rule extensions for real cycles result in a large increase in the number of connectors, summarized in Table ID. Allowing for all possible heat-exchange combinations, the number of possible configurations for cycle n is now 2 Z n (or 2 Z n - l , if the simple cycle is the first one for a refrigerant, since then mixing is not allowed). The total number of configurations for a cascade system with N simple cycles and R refrigerants is 2 ’ t . v + 1 ) - R . For two refrigerants providing five simple cycles, for example, the number of possible configurations becomes 228, or 2.7 x 10s. Identification of Optimal Process Sequence It is clear that even a cascade which is only moderately complex cannot be synthesized by an explicit evaluation of all possible process configurations. However, the stmcture of the problem leads to a simple method of synthesis in which constituent simple cycles are constructed in descending order of refrigerant temperature. The problem of designing a simple cycle is that of finding the optimal path from the initial node to the final node. In order to do so, it is necessary to know the cost of all the possible connections or process operations that may take place. Therefore, all warmer simple cycles with which the vapor stream may exchange energy must already be synthesized, so that the unit cost of the refrigeration produced by them is known. Once the cost of all possible connections is calculated, the minimum-cost path from the initial node of the cycle to the final node is found using a standard dynamic programming approach (Dreyfus, 1969). The cost of the refrigeration produced is calculated and the method is ready to start the synthesis of the next colder simple cycle. The only effects of colder cycles upon the design of warmer cycles have to do with (a) possible subcooling of the condensed refrigerant in warmer cycles, (b) possible superheating of the compressor-inlet vapor of a warmer cycle by mixing with vapor from the next colder simple cycle, and (c) changes in the net flow of refrigerant in warmer cycles, resulting from changes in the refrigeration demand in colder cycles. A vapor-liquid heat exchanger subcooling the liquid in a warmer cycle reduces the net flow of refrigerant in that cycle, whereas a condenser or
Table 111,Process Connections between Network Nodes Rule 3a
Rule 4a
Rule 5a
Rule 5b
Rule 6a
Includes connections f r o m any low -pres s u r e node t o all w a r m e r l o w - p r e s s u r e nodes, representing V-L heat exchange operations. Includes connections f r o m any high-pres s u r e node to all colder high-pressure nodes down t o and including the final node, corresponding t o intercooling, aftercooling, or presaturation. Includes two connections f r o m a lowp r e s s u r e node t o high p r e s s u r e nodes, one representing compression and cooling t o the closest colder high-pressure node, and a second one corresponding t o compression followed by that operation which is optimal for a vapor at the conditions of the closest colder high-pressure node (cooling o r mixing). Eliminates all connections between low p r e s s u r e nodes and high-pressure nodes s t a r t i n g at a t e m p e r a t u r e lower than the minimum allowed f o r the selected m a t e r i a l of construction f o r the c o m p r e s s o r . Includes a connection f r o m any highp r e s s u r e node to the final node, r e p r e senting a mixing operation.
cooler using that refrigerant will increase the required refrigerant flow. Subcooling has no effect upon the design of the vaporcompression portion of the cycle, other than the effect that the change in flow may produce. If more than one vapor is used to subcool a given liquid, the liquid is considered to be split into more than one exchanger in parallel, so that the final temperatures of the liquid effluents from all these exchangers are the same (an equivalent policy would be to allow for multistream heat exchangers). The configurations of the liquid portions of the cycles are fixed by the synthesis of the vapor-handling portions, and hence do not represent an additional problem. Mixing two vapor streams a t a different temperature has the effect of increasing the inlet temperature to the compression stage of the simple cycle where the coldest stream is produced, and therefore increases the cost of that cycle. The problem is handled by assigning this cost increase to the next colder cycle as a cost of mixing, which is estimated by locating the low-pressure node of the warmer cycle closest to the mixed-stream temperature and interpolating or extrapolating the difference between the cost of using that node as the compressor inlet and the cost of using the node selected in the absence of mixing. These costs were obtained during the synthesis of the next warmer cycle. Because of the nonlinearity of dependence of equipment costs upon throughput capacity, refrigerant costs for warmer refrigerants are not known until the demands for refrigerant from colder cycles are known. This difficulty is handled by computing equipment costs for a particular capacity-either an initialized value or the capacity from the previous iteration of the overall procedure-and then linearizing the equipment costs about that value. With this change the refrigerant costs from warmer cycles become independent of refrigeration demands and thereby become iridependent of the structures of colder cycles. A problem arises in that there are at least two ways of
linearizing the refrigeration costs for warmer refrigerants as a function of demand-the average cost given by the total cost of the cycle divided by the total refrigeration demand, and the marginal cost given by the rate of change of cost with refrigeration demand. If only infinitesimal changes in demand are occasioned by changes in structure of colder cycles, then the marginal cost is the correct value to use; however, demand changes occur in finite increments. Usually a modification in the structure of a simple cycle represents only a minor change in the total refrigeration demand of the warmer simple cycles affected by that modification, so that the most reasonable choice is still the marginal cost. This assumption has been borne out by the results of the examples studied. As noted previously, a problem arises when different materials of construction may be used for a particular compressor, depending upon the temperature of the coldest inlet stream. This problem was handled by repeating the synthesis procedure for a given complex cycle once for each candidate material of construction, using Rule 5b to disallow compression operations with inlet temperatures colder than the minimum temperature permitted for that material. The minimum costs obtained for each different compressor material of construction are then compared, and the configuration giving the overall minimum is selected. Iteration upon Process Variables Outer loops are used to update design parameters and optimize the number of levels of each refrigerant employed. Refrigerant Flows. Once a process configuration has been generated, the enthalpy and mass balance equations are solved to calculate the prevailing flows, for which new values of equipment costs and marginal costs are determined. Minimum Temperature Approach. The minimum temperature approach of all heat transfer operations leading to a given node, as well as that for condensation, must be specified initially. In the same loop where refrigerant flows are recalculated, approaches of those heat exchangers which were used in the selected configuration are updated through the use of analytical expressions (Barnes, 1973) based upon the following assumptions: (1) constant specific heats, (2) decoupled variables (cross-effects neglected), and (3) linearized analytical expressions. Experience has shown that these assumptions lead to very good predictions of the actual optimum temperature approaches. The network synthesis procedure is then carried through again, and the entire loop is repeated until convergence is obtained for flows and minimum temperature approaches. Number of Intermediate Levels. One of the most important design variables is the number of intermediate refrigeration levels for each complex cycle corresponding to a single refrigerant fluid. An increase in the number of levels will decrease the cost of compressor work and some capital costs. Other costs, such as piping and insulation, will be increased. The approach taken for defining the number of levels is evolutionary, and consists first of obtaining the optimal configuration for the refrigeration system based on a minimum number of levels specified by the designer for each complex cycle. A heuristic rule is then used to select that complex cycle whose thermodynamic efficiency (ratio of reversible work requirement to actual work requirement) is least, and a new refrigeration level is added to that cycle, with the temperatures of all the intermediate levels Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
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in that cycle being redistributed according to an interpolation formula. The new, converged optimal configuration for the system is obtained. If the cost of the process is not reduced in comparison to the optimal configuration for the previous number of levels, the new level is removed, and that cycle is prevented from further changes. This process is repeated until all complex cycles have been increased to the optimal number of levels. The evolutionary procedure for increasing the number of levels serves as an outermost loop. The interpolation formula used for the temperature distribution of intermediate levels is e = 5 + B ~ ( I- 6)
to prevent any intermediate level from having a temperature outside the allowed range. The results of examples tested show that the cost of the refrigeration system is relatively insensitive to the value of B and that an optimum value is usually obtained for a value of B = 0.5.
Good results were generally obtained when the initial values of the minimum temperature approaches were selected according to the following rules: (a) the initial temperature approach for condensers is 3% of the absolute refrigerant temperature used in the condenser; (b) minimum temperature approaches for V-L heat exchangers and coolers using refrigeration are 25% of the total possible change in the vapor temperature; (c) minimum temperature approach for coolers using water is 3% of the absolute water temperature. An exception to Rule b occurs when a V-L heat exchanger can heat a vapor beyond a transition temperature where a cheaper material of construction is allowed in the compressor. In that case experience showed that the initial temperature approach should be selected so that the exit vapor temperature takes the value of the transition temperature. It should be noted that the first of the heuristic assumptions is also implicit in most methods of synthesis of heat exchanger networks. An exhaustive analysis of the propane cycle (see example below) showed that the second and third heuristic assumptions did not affect the capabilities of the method. However, it is possible that the second assumption could fail with larger mixing effects than those found in the examples explored. If that is the case, one solution would be to modify the synthesis method so that each low-pressure node of any one cycle serves as a candidate final node for the next colder cycle and selection is made of the one that leads to the overall minimum cost. Finally, in the examples analyzed it was found that the functionality of process cost us. number of levels in any one complex cycle has a single minimum independent of the number of levels in other cycles; this result supports the fourth heuristic assumption.
Heuristic Rules a n d Assumptions
Examples
Several rules or assumptions used in the overall synthesis procedure are heuristic in the sense that they have not been proven mathematically to lead necessarily to the global optimum configuration and design. These include the following: (1) the procedure of initially specifying minimum temperature approaches and utilizing linearized expressions for updating minimum temperature approaches for exchangers which have been included in the synthesis will lead to incorporation of all exchangers that should be included and to the optimal minimum temperature approaches of those exchangers; ( 2 ) the best configuration found for a simple cycle in the absence of mixing remains optimal when mixing of compressor-inlet vapors takes place; ( 3 ) the use of linearized marginal costs of warmer refrigerants will lead to the optimal configuration despite the fact that changes in process configuration result in finite changes in refrigeration demands in warmer cycles; (4) the method used for locating the optimal number of levels for each complex cycle leads to the global rather than a local optimum. Numerous examples were studied to determine the validity of these assumptions ( B a r d s , 1973). The first assumption was tested by starting a given synthesis problem with quite different values of initially specified minimum temperature approaches. All the results showed that the linearized expressions used to update the minimum temperature approaches lead very rapidly to the converged value. However, in some cases it was found that, when different initial values were stipulated, the method could lead to nonoptimal configurations showing local optimum conditions with respect to the values of minimum temperature approaches.
Several examples were studied to test the capabilities of the method to select the best cycle configuration and to explore the effects of different design variables upon the optimum configuration and number of levels (Barn&, 1973). Some of these examples are discussed here. Propane Refrigeration. The first case analyzed is a step demand for 5000 kW of refrigeration a t 235 K, which could correspond to a demand arising from the condensation of a narrow-boiling mixture or a single component. Figure 5 shows the effect of the number of levels and the temperature distribution parameter B on the unit cost of the refrigeration obtained. The optimal configuration with N = 2 and B = 0.5 is shown in Figure 6. The results in Figure 5 show that there is a small incentive for using one economizer, but that the cost would increase if a second one is added. The results also show that there is only a weak dependence of the process cost upon temperature distribution and that a minimum is obtained when B has a value of about 0.5. When B has a value 0.8 or greater there is a change in the optimal configuration for two levels, represented by the broken line in Figure 5 . With higher values of B, the compressed vapor from the coldest level is presaturated before further compression. In that case the configuration with the presaturator consumes more energy than the configuration without it, but it has the advantage that the condenser can now remove a larger proportion of the heat a t a lower unit cost than in the cooler. This is because of competing effects of the heat transfer coefficient and the mean temperature approach in determining which exchanger has the lowest capital cost per unit of heat removed. It is also interesting to note that the optimal configura-
where l / T i - I/T, 1/T, - 1/T,
e= 5
.
.
z - a, =-
N
where i is the number of the refrigeration level, i, is the number of the coldest level in the complex cycle, T, is the temperature of the coldest level, and Tc is the temperature of the refrigerant used in the condenser. The parameter B is selected by the designer, so that IBI
428
N < N-1
(6)
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*'K Cost
(S/KW-yr)
145.
Tv* 2 3 5 K
Ov= 5000 KW
