8724
Ind. Eng. Chem. Res. 2008, 47, 8724–8740
Optimal Design of Mixed Refrigerant Cycles
Frank Del Nogal, Jin-Kuk Kim,* Simon Perry, and Robin Smith
Centre for Process Integration, School of Chemical Engineering and Analytical Science, The UniVersity of Manchester, PO Box 88, Manchester, M60 1QD, U.K.
A new approach for the optimal design of mixed refrigerant cycles is presented. It is based on mathematical programming and offers signi?cant improvements in relation to previous approaches. It includes multistage refrigerant compression, full enforcement of the minimum temperature difference in heat exchangers, simultaneous optimization of variables, consideration of capital costs, and the use of stochastic optimization (genetic algorithm) to overcome local optima. The approach can be applied to either single mixed refrigerant cycles or to systems consisting of two of these in cascade. The effectiveness of the method is illustrated by revisiting previously published liqui?ed natural gas case studies, for which better and feasible solutions are produced, and which prove the importance of considering multistage compression and capital costs during optimization. The application of genetic algorithms in the design of mixed refrigerant cycles permits a greater con?dence in the optimality of the results.
1. Introduction Most low temperature processes feature one or more refrigeration cycles with the purpose of removing heat from subambient hot streams. The provision of a cryogenic cooling requires signi?cant power demands for compression, and it is very important to achieve high energy ef?ciency in the design and operation of refrigeration cycles, leading to low carbon emissions to the environment. In a simple system with a closed refrigeration cycle, the heat is removed by vaporization of a low pressure refrigerant which is then compressed and condensed at a higher pressure against a warmer cold utility or heat sink. The condensed liquid is let down in pressure (and temperature) by means of an expanding device such as a throttle valve. When cooling for a wide temperature range is required, a complex arrangement (for example, a cascade cycle or a cycle with multilevel cooling) is introduced to improve thermodynamic ef?ciency of the refrigeration systems. On the other hand, using mixed refrigerant (MR) in the refrigeration cycles provides very promising potential to yield more ef?cient, yet simple and reliable systems in comparison to pure refrigerant ones, because a mixture of refrigerants is evaporated isobarically, not at a single but in a range of temperatures. Although there are important “natural” applications for MR cycles (e.g., LNG), their optimal design has not been the object of extensive research as in the case of pure refrigerant systems. Therefore, to explore advantages using mixed refrigerants in low temperature cooling, it is the aim of this paper to develop a systematic design and synthesis optimization framework for MR systems. The new design methodology provides systematic investigation of design interactions as well as optimal and economic design of MR systems. First, a brief overview is made regarding the design problems being tackled in the area of mixed refrigeration systems, which is followed by the review of existing literature as relevant to the optimal design of MR systems, identifying improvement opportunities and gaps in the existing design practice. Next, the optimal design of MR systems is presented, in which the
* To whom correspondence should be addressed. E-mail: j.kim-2@ manchester.ac.uk. Tel.: +44 (0)161 306 8755. Fax: +44 (0)161 236 7439.
details of the mathematical formulation and optimization strategy are provided for both single MR systems and MR systems in cascade, including multistage compression and typical ?owsheet/ equipment variations. Finally, the proposed approach is illustrated with two case studies. 2. MR Systems 2.1. Pure Refrigerant and MR Cycles. A major limitation of vapor compression cycles that make use of pure components as refrigerants is that refrigeration is provided at a constant temperature while the cold refrigerant is evaporating. For a chosen refrigerant, the temperature at which cooling is provided is a consequence of the evaporator pressure, that is, the saturation temperature. If the hot streams (streams that need cooling) demand the cooling task to be carried out along a wide temperature range, a system providing all the refrigeration at a single level is likely to have a poor performance. This is because large temperature differences would exist in the heat exchanger(s), moving the system away from thermodynamic reversibility, and hence, from thermodynamic ef?ciency. That being the case, a multilevel pure refrigerant system is likely to be implemented, in an attempt to reduce the temperature differences in the system, as seen in Figure 1, in which three pressure levels are used to provide refrigeration at three temperature levels. However, both heat transfer area and complexity would increase as a consequence. The refrigerant compressor would need as many stages as refrigeration levels. Multicomponent, or mixed refrigerants, unlike pure refrigerants, undergo isobaric phase change through a range of temperatures contained within the dew and bubble temperatures of the mixture. Given the right pressures and compositions, a good match between the process and refrigerant temperature pro?les can be obtained with a simple con?guration, as shown in Figure 2. Although refrigerant compression can take place in several compression stages with intercooling, only one stage is shown in the illustration for simplicity. A practical maximum pressure ratio per compression stage (usually 4-5 for industrial applications) is likely to de?ne the number of compression stages, rather than the number of pressure levels, since in this case the cold refrigerant evaporates at a unique pressure (or through a small pressure range, if friction pressure drop is
10.1021/ie800515u CCC: $40.75 ? 2008 American Chemical Society Published on Web 10/18/2008
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8725
Figure 1. A three-level pure refrigerant cycle.
Figure 2. A MR cycle.
Figure 3. A self-cooling MR cycle.
accounted for in the evaporator). Care should be taken, however, since the reduced average temperature difference would lead to increased heat transfer area. A minimum temperature difference (?T min) should be established for practical design purposes. Since the bene?ts of using MR cycles are highlighted in situations where refrigeration is to be provided along wide temperature ranges, these types of cycle ?nd a natural application in the liqui?ed natural gas (LNG) industry, where refrigeration is required from ambient temperature to around -160 °C. 2.2. Flowsheet Variations of MR Cycles. An interesting ?owsheet variation in MR cycles is the one shown in Figure 3, where the high pressure refrigerant rejects heat not only to an
Figure 4. A three-stage MR cycle.
external heat sink but also to itself, once expanded. In this way the hot refrigerant is further cooled down, probably subcooled,
8726 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 5. MR structure used by Vaidyaraman and Maranas.6
Figure 6. MR system diagram.
before expanding, allowing the cold refrigerant to reach a lower temperature and/or a lower vapor fraction after the expansion. Such a bene?t is achieved at the expense of a higher heat transfer area, since the total duty in the heat exchanger increases signi?cantly. A multistream heat exchanger can be used to handle all hot streams and the cold stream in a single piece of
equipment. This process is known as the Pritchard cycle and is described in more detail by Walsh.1 Repeated partial condensation and separation of the refrigerant stream has been reported to achieve a better match between the temperature pro?les.2 However, this should not be taken as a general rule. Figure 4 shows a MR cycle with three refrigeration
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8727
Figure 7. Temperature feasibility check.
stages, as an extension of Figure 3a. Although increasing the number of stages may reduce the power consumption, the design also grows in complexity and probably in capital cost. This results in an important tradeoff for designers. 2.3. Multistage Compression. As compressors used in industry have a practical maximum stage pressure ratio of around 4-5, it is very common to ?nd compression tasks performed in multiple stages. Further, if there is an appropriate cold utility, the partially compressed gas originating from a given stage may be cooled down before entering the next compression stage (intercooling). The lower temperature of the partially compressed gas reduces the volumetric ?owrate and in consequence reduces the compression power of the next stage. If part of the gas condenses, then it is necessary to remove the liquid so that a dry vapor stream enters the next compression stage. The liquid removed is then pumped and mixed with the fully compressed gas and with any liquid streams from other intercoolers at the outlet of the last compression stage at same pressure. Condensate removal and pumping also helps in saving power, as increasing the pressure of liquids is much cheaper in power (and capital) than doing so with gases. 3. Literature Review Although there are many publications on the analysis of MR systems, only a few exist on their optimal design using mathematical programming. The earliest attempt to optimize these systems in a systematic manner was carried out by AitAli.3 This work tackled the optimization of an MR system with the same con?guration as a two-level pure refrigerant system. The focus was largely on minimizing total compression power by trying to enforce a constant temperature difference through the cryogenic heat exchangers. Although rich in practical considerations and insights relevant to LNG production, refrigerant pressures and ?owrates were not the result of optimization, but rather set heuristically. Different solution procedures were required depending on the number of components present in the refrigerant. Also, to handle the complexity of the problem, only binary and tertiary mixtures were considered and the thermodynamic accuracy had to be sacri?ced by using an equation of state based on ideal solution assumptions and Raoult’s law for vapor-liquid equilibrium, which restrict the validity of the numerical results to low pressure and warm temperature conditions. The optimization method was limited to a two-dimensional numerical search. Refrigerant subcooling and separation of liquid and vapor refrigerant streams at
intermediate temperatures were not addressed, neither was cascading heat to a different refrigeration cycle. Lee4 (part of this work was also published as Lee et al.5) worked on the optimal design of multistage MR cycles. Given a refrigeration task in the form of a hot composite curve (a combined temperature-enthalpy pro?le of all hot streams) and the number of refrigeration stages, his approach allows for the optimization of key variables in a process ?owsheet of the type shown in Figure 4. In principle, the optimization variables are the refrigerant composition and ?owrate (at compressor inlet), the compressor inlet and outlet pressures, and the intermediate temperatures (at which refrigerant is separated into vapor and liquid streams). However, the decision variables are not optimized simultaneously. Instead, the refrigerant compositions are optimized at ?xed refrigerant ?owrate and pressures using nonlinear programming (NLP). Once the optimal composition is obtained, the hot and cold temperature pro?les are checked for feasibility. If they do not cross, then new refrigerant ?owrate and pressures are proposed on the basis of heuristics, judgment, or optimization. And the procedure is repeated until no further improvementispossiblewithoutincurringtemperatureinfeasibilities. Lee4 also used three different types of objective function. Two of them tried to match the temperature pro?le of the cold refrigerant to an ideal pro?le (the hot composite curve shifted down by ?Tmin) by minimizing either the maximum violation of the minimum temperature difference or the sum of such violations along the pro?le. The third possible objective function was the minimization of compressor power, but its use was not recommended until the ?nal part of the overall optimization. That is because, since there is no constraint on the vapor fraction at the inlet of the compressor, it is less likely to have wetness at the compressor inlet toward the end of the overall optimization due to a probable lower value of the refrigerant ?owrate. The design of MR cycles is a highly nonlinear problem with many local optima. One of the main drawbacks of Lee’s approach is that the nonsimultaneous optimization of variables coupled with the dependence of the ?nal solution on the initial guess, on the nonsystematic selection of refrigerant ?owrate and pressures (especially if updated using heuristics or judgment) and on the switching of the objective function, makes reaching the global optimal solution (or a good near-optimal solution, for practical purposes) very unlikely. In Lee’s method, the minimum temperature difference between the hot and cold temperature pro?les is not fully enforced, which often leads to very tight pro?les, apart from relying on human intervention at each iteration to check for
8728 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
temperature feasibility. It is assumed that the refrigerant compression takes place in one stage with no intercooling, regardless of the compression ratio. No capital costs are considered during the optimization procedure. Lee4 also explored the synthesis of more complex systems such as MR cycles in cascade with pure refrigerant cycles. In his approach, although the partition temperature between the two cycles was subject to optimization, each cycle was optimized separately to minimize compression power for an assumed value of partition temperature. The partition temperature was then iterated manually in an outer loop until the total power was considered optimal. Again, this nonsimultaneous, nonsystematic optimization of variables is likely to lead to nonoptimal solutions, as discussed before. Another relevant precedent in the synthesis of multistage MR cycles is that by Vaidyaraman and Maranas,6 which is also based on NLP. The cycle topology used by them is the one shown in Figure 5a. Although in principle it may look very different, it is just a slight variation of the type of structure in Figure 4, only that the liquid refrigerant stream is not subcooled after each ?ash step, as shown in Figure 5b as an alternative representation. The variables being optimized in this case were the refrigerant composition (at the stream being expanded in the last stage), the compressor inlet and outlet pressures and the vapor fraction at ?ash drums 2 to N. All the design variables (each with a ?xed number of stages) were optimized simultaneously. Two key assumptions were made by Vaidyaraman and Maranas.6 The ?rst one was that the hot refrigerant leaving stage N was at its bubble point, and the second one was that the cold refrigerant streams leaving each stage were at their dew point. These assumptions, although not unreasonable, constrain the solution space unnecessarily and could lead to good opportunities missed by the optimizer. Another major shortcoming of the method proposed is that temperature feasibility is only enforced at the ends of the heat exchangers. This opens the possibility of not only minimum temperature difference violations but also of temperature crossovers being overlooked during the optimization. Although the authors suggest that, if violated, feasibility might be regained after the optimization by correcting the refrigerant pressures, doing so could lead the solution to lose its optimality. As in Lee,4 Vaidyaraman and Maranas6 assumed single stage compression without intercooling and did not consider capital costs. However, they made a further effort in trying to overcome local optima by performing a number of optimizations with different starting points. The effect of refrigerant pressure drops was not covered in their formulation. In overall, the review of previous work on MR systems reveals a quite signi?cant potential for improvement. An ideal design approach would be one that combines (a) ?exibility to handle ?owsheet options such as multiple refrigeration stages, refrigerant subcooling, and cascade systems, (b) energy-ef?cient features, such as multistage compression, (c) a systematic optimization framework powerful enough to exploit the relevant degrees of freedom and to overcome the challenges of a highly nonlinear problem, (d) no assumptions on the thermodynamic state of refrigerant streams just for the sake of a simpli?ed calculation procedure, (e) full feasibility enforcement, and (f) the cost consequences of the design decisions. A design approach like this is not yet available in the open literature, and in order to overcome shortcomings addressed in above, a novel, systematic and
robust procedure for the optimal design of low temperature processes is proposed in the next section. 4. Design and Optimization Frameworks Figure 6 shows a generic multistage MR cycle and a multistage compression system with intercooling, indicating some of the nomenclature used in the formulation. Variables for the hypothetical refrigeration stage 0 and compression stage 0 were included in order to simplify the formulation. Also, for the sake of a compact formulation, some calculations are not described explicitly. These are represented as functions of the type f(x1,..., xN) and are mainly routine physical property calculations (e.g., enthalpy for a given set of compositions, temperature, and pressure) and temperature pro?le operations. It is assumed that all hot streams leaving each multistream heat exchanger are at the same temperature. Objective Function. Equation 1 represents the objective function as a generic function of the main process variables. This objective function may well change from case to case according to the purpose of the designer (e.g., minimum compressor power, minimum capital investment, minimum total cost, etc.) and to the economic models used. OBJECTIVE ) OBJ(WC, QCMP, THCOMP, HHCOMP, TCCOMP, HCCOMP, ...) (1) Stage Material and Energy Balances. Equations 2-7 establish the relationship between the total ?ows of the streams around the system. Equations 2-4 describe the continuity of vapor and liquid hot refrigerant stream ?ows from stage to stage. The hot vapor refrigerant stream is made unavailable at the NRth stage because of the inexistence of a ?ash drum at stage NR-1 (eq 5). All the material arriving to stage NR from stage NR -1 is put through the hot refrigerant liquid stream for formulation purposes, although part of it might not be in the liquid state. Equations 6 and 7 represent the material balance of the mixers. Fn ) FVn + FLn Fn ) FVn-1
OUT FVn ) VF(Yn-1, TIn-1, PHn -1 )
(2) (3) n e NR - 1 (4) (5)
FVNR ) 0 FCn ) FLn + FCn+1 FCNR ) FNR n e NR - 1
(6) (7)
Vapor and liquid compositions are obtained from phase equilibrium calculations at each ?ash drum (eq 8 and 9) although in practice the vapor and liquid compositions are obtained simultaneously from a single ?ash subroutine. The composition of the only hot refrigerant stream in stage NR must be the same as that of the hot vapor refrigerant at the previous stage (eq 10). Equation 11 states that the total composition of the cold refrigerant at each stage is the same as that of the hot vapor refrigerant at the previous stage and is derived from component balances carried out around the ?rst n - 1 stages as a group.
OUT Yn ) ZVAP(Yn-1, TIn-1, PHn -1 ) OUT Xn ) ZLIQ(Yn-1, TIn-1, PHn -1 )
(8) (9) (10) (11)
n e NR - 1
XNR ) YNR-1 Zn ) Yn-1
The heat removed from hot refrigerant streams at each stage is calculated in eq 12 and 13. The initial and ?nal values of the
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8729
vector of intermediate temperatures (TI) are set to TR and TPOUT, respectively, for consistency with the input data (eqs 14 and 15), although TI0 and TINR are not actual intermediate temperatures. Since the refrigeration task is assumed to be given in the form of a precalculated process composite curve (PR, HPR), eq 16 calculates the heat removed from the process stream(s) at each stage according to the enthalpy difference of such a curve between the respective pair of adjacent intermediate temperatures, the exception being stage 1 (eq 17), because in principle the initial process temperature (TPIN) might be different from the initial hot refrigerant temperature (TI0 ) TRIN). Equation 18 ensures that the total heat removed from the hot streams is absorbed by the cold refrigerant stream at each stage. As a result, the cold refrigerant outlet enthalpies and temperatures can be determined as a result of an energy balance (eqs 19 and 20).
OUT QVn ) FVn[hTP(Yn, TIn, PHOUT ) - hTP(Yn, TIn-1, PHn n -1 )] (12) OUT QLn ) FLn[hTP(Xn, TIn, PHOUT ) - hTP(Xn, TIn-1, PHn n -1 )] (13)
IN
at the inlet of stage NR, the inlet cold refrigerant temperature is the same as the one after expansion (eq 27).
IN hCn ) EXPN IN OUT IN FLn · hTP(Xn, Tn , PLn ) + FCn+1hTP(Zn+1, TCn +1 , PLn ) × FCn
n e NR - 1 (25)
TCIN n ) ThP
(
IN Zn, hCIN n , PLn
)
n e NR - 1
(26) (27)
IN EXPN TCNR ) TNR
TI0 ) TRIN TINR ) TPOUT QPn ) EVAL(TPR, HPR, TIn) EVAL(TPR, HPR, TIn-1)
(14) (15)
n g 2 (16)
QP1 ) EVAL(TPR, HPR, TI1) - EVAL(TPR, HPR, TPIN) (17) QCn ) - (QVn + QLn + QPn) ) hCOUT n FCnhTP(
IN Zn, TCIN n, PLn
Pressure, Temperature, And Enthalpy Pro?les. Accurate pressure drop predictions across heat exchangers, especially of the compact type, may result in a quite lengthy calculation task, since details of the actual exchanger geometry and internals (e.g., type of ?ns), as well as a set of additional ?uid transport properties, are required for such a purpose.7 If a rigorous calculation of such kind was implemented in the present formulation it would increase the mathematical complexity considerably, taking the focus away form the main design variables. On the other hand, neglecting the pressure drops in the cycle could lead to overlooked infeasibilities because of an inaccurate prediction of temperature pro?les and/or to a nonoptimal and/or subdesigned system. In the present formulation, an intermediate point is adopted. The total hot refrigerant OUT OUT pressure drop (PHNR -PH0 ) has a ?xed value de?ned by the user before the optimization (?PHOT), which allows the estimation of intermediate hot refrigerant pressures by approximating them as a linear function of the respective temperature, as shown in eq 28. A similar approximation is performed to obtain the intermediate cold refrigerant pressures (eq 29). However, these are assumed as a linear function of the temperature of the hot refrigerant, instead of that of the cold refrigerant for convenience in the sequential simulation approach. ) PHOUT PHOUT n 0 TIn - TRIN TP
IN PLIN n ) PL0 + OUT
(18) (19) (20) - TR
IN
) + QCn
?PHOT
(28)
FCn
IN TCOUT ) ThP(hCOUT , PLn n n -1)
The enthalpy of the refrigerant after expansion is calculated in eq 21. Depending on the value of the user de?ned binary parameter YLEX, such an enthalpy will be set equal to either the inlet enthalpy (for YLEX ) 0, corresponding to JouleThompson valves) or to a function of the isentropic enthalpy of expansion and a constant isentropic ef?ciency (for YLEX ) 1, corresponding to liquid expanders). The enthalpy of isentropic expansion is calculated in eq 22. The refrigerant temperature after expansion is given in eq 23. When expanders are used, eq 24 will calculate the associated power produced. Otherwise this power would be implicitly forced to zero as in such case there is no enthalpy difference. ) (1 - YLEX)hTP(Xn, TIn, PHOUT hEXPN )+ n n YLEX(1 - ηLEX)hTP(Xn, TIn, PHOUT (21) ) + ηLEXhISLEX n n ) hsP(Xn, sTP(Xn, TIn, PHOUT hISLEX ), PLIN n n n ) TEXPN ) ThP(Xn, hEXPN , PLIN n n n ) - hTP(Xn, TIn, PHOUT WLEXn ) FLn(hEXPN )) n n (22) (23) (24)
(29) TP - TR For a given composition and inlet and outlet temperatures, the temperature-enthalpy pro?les at each refrigeration stage are obtained by evaluating the stream enthalpy repeatedly at a number of intermediate points, previously de?ned by the designer (i.e., the number of points, not the points themselves), within the respective temperature range. A linear behavior of refrigerant pressure with temperature is also assumed during the calculation of the temperature-enthalpy pro?les. Cold refrigerant pressures, this time, are estimated according to the cold refrigerant temperatures and not those of the hot refrigerant as done previously. Extra evaluation points are added at the exact dew and/or bubble points of the mixture if found within the temperature and pressure range. The functions TPROF and HPROF are in practice one single function being shown separately for formulation purposes. The hot vapor refrigerant, hot liquid refrigerant, and cold refrigerant pro?les are de?ned in eq 30 to 35.
OUT IN OUT OUT , Yn) THVn ) TPROF(TIn-1, TIn, PHn -1 , PHn
TIn - TRIN
?PCOLD
n e NR - 1 (30) HHVn ) HPROF(
OUT OUT TIn-1, TIn, PHn , Yn, FVn -1 , PHn
The inlet cold refrigerant temperature to any given stage featuring an inlet mixer (stages 1 to NR -1) can be calculated (Equation 26) after its enthalpy is known through an energy balance around the mixer (Equation 25). Since there is no mixer
) )
(32)
n e NR - 1 (31) THLn ) TPROF(
OUT OUT TIn-1, TIn, PHn , Xn -1 , PHn
8730 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
OUT OUT HHLn ) HPROF(TIn-1, TIn, PHn , Xn, FLn) (33) -1 , PHn OUT IN TCn ) TPROF(TCIN , PLIN n , TCn n , PLn-1, Zn)
, TCCOMP , HEV(THCOMP , HHCOMP , th)) g th - TEV(HCCOMP n n n n ?Tmin ? th ∈ THCOMP (46) n Compression. In practice, stages belonging to the same compressor feature pressure ratios not too different from each other. Identical pressure ratios are therefore not an unreasonable assumption. However, allowing dissimilar pressure ratios would mean more degrees of freedom during optimization, which may bring the advantage of an even further improvement of the objective function value (e.g., total compression power) and/or a more ?exible matching with mechanical drivers. The formulation in this work is ?exible and allows both approaches; dissimilar compression ratios subject to optimization when full exploitation of the degrees of freedom is required and an identical compression ratio approach in case a simpler optimization is desired. Equation 47 ensures that the values of stage pressure ratios are consistent with the compressor inlet and outlet pressures. The discharge pressures of each compression stage are then calculated by eq 48. Equations 49 and 50 enforce minimum and maximum limits to the stage pressure ratios. In case the designer would like to perform the optimization assuming identical pressure ratios, the unique variable PR would replace the set of variables PRi in the problem formulation. Equation 51 states that the number of stages is such that the compression task would not be possible with one stage less. Although this may appear redundant with eq 47, it is necessary for the identical pressure ratio approach since in that case the unique variable PR is not controlled by the optimizer. As a consequence, there may be more that one set of NC and PR values that satisfy eqs 47, 49, and 50 simultaneously, and to ensure the number of stages is not excessive, eq 51 has to exist in order to de?ne the minimum number of stages as the design criteria in that situation. PLIN 0
(34)
OUT IN HCn ) HPROF(TCIN , PLIN n , TCn n , PLn-1, Zn, FCn) (35)
Since the overall process composite curve is given as an input to the problem, the process pro?les at each stage are obtained by just extracting the corresponding data within each respective range of temperatures. The overall process pro?le input to the problem should consider the effect of pressure drop. The process data extraction is performed in eq 36-39. If the process composite curve was not an input to the problem, the process pro?les could be constructed using a similar method as for the refrigerant pro?les. TPn ) CROP(TPR, TIn-1, TIn) TP1 ) CROP(TPR, TPIN, TI1) HPn ) CROP(HPR, TIn-1, TIn) HP1 ) CROP(HPR, TPIN, TI1) ng2 ng2 (36) (37) (38) (39)
Equations 40-43 combine all the hot pro?les to form a single hot composite curve in each stage. Equations 44 and 45 obtain the stage cold refrigerant composite curves. ) TCOMP(THVn, THLn, TPn, HHVn, HHLn, HPn) THCOMP n n e NR - 1 (40) TH COMP ) TCOMP(THLNR, TPNR, HHLNR, HPNR) (41) NR ne
HHCOMP ) n HCOMP(THVn, THLn, TPn, HHVn, HHLn, HPn)
NR - 1(42) COMP ) HCOMP(THLNRTPNR, HHLNR, HPNR) (43) HH NR TCCOMP ) TCOMP(TCn, HCn) n HCCOMP ) HCOMP(TCn, HCn) n (44) (45)
∏ PR ) PH
j)1 j i j)1
NC
OUT 0
(47) (48) (49) (50)
) PLIN PCMP i 0
∏ PR
j
Once the hot and cold composite curves are known at each stage, the temperature feasibility at each stage can be evaluated by comparing the temperature of the hot composite curve to that of the cold one for a given enthalpy. Such comparison is performed at each enthalpy point belonging to the hot composite curve. If the temperature of the cold composite evaluated at a given enthalpy is greater than that of the hot composite then a temperature cross occurs. If the temperature of the cold composite is lower than that of the hot composite by less than ?Tmin then a violation of the minimum temperature difference occurs. If the temperature of the cold composite is lower than that of the hot composite by at least ?Tmin then the point is perfectly feasible. This temperature feasibility check is performed by eq 46 and further illustrated in Figure 7. Checking feasibility for a couple of points only will not ensure overall feasibility for heat transfer throughout overall temperature range, whereas employing a large number of discrete points with very small interval will signi?cantly increase computational time in the optimization. Various numbers of points have been tested, and in this study, around 30 points (together with extra evaluation points, depending on the characteristics of composite curves) were taken, which was enough to check feasibility of heat recovery in exchangers without using excessive computational resources.
PRi e PRMAX PRi g PRMIN > PLIN PHOUT 0 0
NC-1 j)1
∏ PR
j
(51)
Equation 52 avoids any wetness in the cold refrigerant stream from refrigeration stage 1, which is the stream entering the compression train. Equations 53 and 54 make the compression nomenclature consistent with that of the remainder of the cycle. g TDEW(Y0, PLIN TCOUT 1 0 ) PCMP ) PLIN 0 0 YCMP ) Y0 0 (52) (53) (54)
Equations 55-59 establish refrigerant ?owrates and compositions around the compression system in consistency with phase equilibrium and stage material balances. It is assumed that all the partially compressed streams can be cooled down to a temperature TRIN by an external heat sink and that any liquid that is formed is separated, pumped to the ?nal
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8731
pressure, and remixed before the ?nal cooler, in accordance with Figure 6b. ) ZVAP YCMP i
OUT WLEXn ) FLn(hEXPN - hTP(Xn, TIn-1, PHn n -1 ))
(24b)
(
CMP IN YiCMP -1 , TR , Pi
) )
i e NC - 1 (55)
THCOMP ) TCOMP(THVn, TPn, HHVn, HPn) n n e NR - 1 (40b) ) HCOMP(THVn, TPn, HHVn, HPn) HHCOMP n n e NR - 1 (42b) 5. Solution Strategy The optimization problem, as formulated in the previous section, is of the NLP type: Minimize eq 1, subject to constraints 2-64 (with variations 13b, 21b, 29b, 30b, and 37b-40b, if appropriate). However, not all the formulation has been explicit (e.g., phase equilibrium and physical property calculations), and in practice part of the calculations may take place in external subroutines (e.g., interfacing with commercial process simulators). Also, as stated before, the problem is nonlinear and features many local optima. Traditional deterministic optimization methods would get readily trapped in these. On the other hand, stochastic methods (e.g., genetic algorithm, simulated annealing) may offer more con?dence on the optimality of the ?nal solution at the expense of computational time. A genetic algorithm (GA) has been chosen as the main optimizer in this work. Genetic algorithms try to copy nature regarding the evolution of species. They perform iterations not on a single candidate solution but on a population of candidate solutions (individuals) instead. The path toward optimality is built by sharing information between individuals (reproduction) and by evaluating and ranking the resulting offspring according to their relative optimality (?tness) in a given number of major iterations (generations). The information de?ning a particular individual (chromosome) consists of a set of values of the chosen independent variables (genes), over a discretized solution space, encoded as a string (e.g., binary encoding, real encoding). Another key advantage of GA is that no initial guess is needed to start up the optimization process. Further details on how genetic algorithms work can be found in books such as Goldberg8 and Stender et al.9 Pikaia,10 a genetic algorithm by the U.S. National Center for Atmospheric Research, has been adopted in this work and modi?ed in order to improve the quality of the initial generation. Randomly generated individuals are now evaluated before being allowed into the initial generation. They qualify as a member of the initial generation only if excessive infeasibilities are not incurred (e.g., average crossovers of no more than 3 °C and a compressor inlet temperature at most 2 °C below dew point), otherwise they are rejected. This ?lter, although it increases the computational time for the ?rst generation considerably, ensures a departure from a better set of candidate solutions, shortening the path to optimality. For case studies in this paper, 90?95% of candidates are rejected during the ?ltration step. Figure 8 illustrates the ?ow of information within the general optimization framework. The optimizer (GA in this case) proposes a series of candidate solutions and relies on the simulator for their assessment. How the optimization task evolves is decided by the optimizer on the basis of the assessment of the candidate solutions. The interactions between the GA and the simulator result in a set of the best solutions found over a discretized solution space. Standard NLP optimization(s) can be carried out afterward having the best discretized solution(s) as initial guess in order to ?ne-tune and ?nally report the optimal solution on the basis of a continuous solution space. Both the optimizer and the simulator have been implemented
CMP IN XCMP ) ZVAP YiCMP i -1 , TR , Pi CMP CMP IN ) FiCMP FCMP i -1 VF Yi-2 , TR , Pi-1
(
)
ng2
(56) (57) (58)
(
FCMP ) F1 1
CMP FP - FiCMP i ) Fi +1
i e NC - 1
(59)
Stage compression power is calculated in eqs 60 and 61on the basis of the stage ?owrate and enthalpy difference. Stage outlet enthalpy is calculated using a constant isentropic ef?ciency in eq 62 and 63, the last one being derived from the de?nition of isentropic ef?ciency. Stage pumping power is calculated in a similar way by eqs 64-66
CMP IN (hCMP - hTP YiCMP WCi ) FCMP i i -1 , TR , Pi-1
(
)
ng2 (60) (61)
· (hCMP - hTP(YCMP , TCOUT , PCMP WC1 ) FCMP )) 1 1 0 1 0
CMP CMP CMP IN hISCMP ) hSP YiCMP i -1 , sTP Yi-1 , TR , Pi-1 , Pi
(
(
)
)
(62)
CMP CMP IN hCMP ) (1 - ηc)hTP YiCMP (63) i -1 , TR , Pi-1 + ηchISi CMP WPi ) FPi hP , TRIN, PCMP i - hTP Xi i
(
)
(
(
))
i e NC - 1 (64)
CMP , TRIN, PCMP + ηphISP hP i ) (1 - ηp)hTP Xi i i
(
)
i e NC - 1 (65)
CMP hISP , sTP XCMP , TRIN, PCMP , PHOUT i ) hsP Xi i i 0 i e NC - 1 (66) Finally, eq 67 calculates the heat removed at each intercooler except for the one after the last compression stage, which is calculated in eq 68 based on an energy balance around the last intercooler and all the mixers in conjunction. CMP IN ) FCMP · hTP YiCMP - hCMP QCMP i i -1 , TR , Pi i
(
(
)
)
( (
)
)
i e NC - 1 (67)
CMP QNC ) F1hTP(ZCMP , TRIN, PH) i
∑
NC-1 i)1
P CMP CMP (hP i · Fi ) - FNC hNC (68)
Adaptation for Hot Liquid Refrigerant without Subcooling. MR cycles without hot liquid refrigerant subcooling, as in Figure 5, are a particular case in the formulation above. If such a cycle was to be represented, some constraints would change: Namely, all QLs would be equal to zero; the refrigerant OUT expansions would start at conditions TIn-1 and PHn -1 , instead OUT of TIn and PHn ; there would not be need for calculating any hot liquid refrigerant pro?le (i.e., eqs 32, 33, 41, and 43). The equations to be modi?ed are shown below as variations b of the original equations: QLn ) 0
OUT hEXPN ) (1 - YLEX)hTP(Xn, TIn-1, PHn n -1 ) + OUT LEX YLEX(1 - ηLEX)hTP(Xn, TIn-1, PHn (21b) -1 ) + ηLEXhISn OUT IN hISLEX ) hsP(Xn, sTP(Xn, TIn-1, PHn n -1 ), PLn )
(13b)
(22b)
8732 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 8. Optimization strategy. Table 1. Temperature-Enthalpy Data of the Natural Gas Stream to Be Lique?ed temperature (°C) 25.0 -6.0 -34.1 -57.7 -70.1 -74.6 -82.3 -96.5 -115.0 -163.0 enthalpy (MW) 20179 18317 16353 14468 11978 10198 7114 5690 3840 0
Figure 9. Composite curves for the base case optimal solution. Table 2. The Three Solutions for the Base Case new method ?Tmin (°C) power (MW) F1 (kmol/s) IN PL0 (bar) OUT PH0 (bar) Z1 (mol %) N2 CH4 C2H6 C3H8 n-C4H10 5.0 33.49 3.47 2.40 36.95 15.32 17.79 40.85 0.41 25.62 Lee’s method 1.2 26.60 3.2 3.7 40.0 11.0 27.3 35.6 5.20 20.9 new method 1.2 24.53 3.53 4.84 43.87 10.08 27.12 37.21 0.27 25.31
in FORTRAN 77 and within WORK, part of the process design software suite available from the Centre for Process Integration at the University of Manchester. This gives the user the choice of using either built-in thermodynamic property models or delegating these calculations to commercial process simulators such as HYSYS or Aspen Properties that would run in the background. The chosen independent variables for this application are the refrigerant ?owrate (F1) and composition (Z1), the compressor inlet and outlet pressures (PHOUT and PLIN 0 0 ) and the intermediate temperatures (TI1 to TINR -1) for cycles with two stages or more. Given the values for all these variables, the simulator is responsible for calculating the remaining variables and the magnitude of infeasibilities (e.g., temperature pro?les and compressor inlet wetness) and evaluating the objective function. Penalties may be applied to the objective function according to the magnitude of the infeasibilities as a disincentive in the ranking of solutions incurring violations. However, during optimization, care should be taken in removing design candidates with considerable magnitude of infeasibilities in the objective function since infeasible solutions (especially the slightly and moderately infeasible) may still contain features worthy to inherit and evolve by offspring candidate solutions. 6. Case Study 1 The problem formulation and optimization strategy discussed in the previous section are put into practice using the same LNG case study published by Lee.4 A pretreated natural gas stream is to be cooled from 25 °C down to -163 °C using a mixture of nitrogen, methane, ethane, propane, and n-butane as refrigerant in a single stage cycle (as in Figure 3a) using minimum compression power as the objective function. Subcooling of liquid refrigerant is allowed and compression takes place in one stage. External cold utility is available to cool hot streams down to 30 °C. The minimum temperature difference is 5 °C, the
isentropic compression ef?ciency is 80%, and the physical properties calculations are based on the Peng-Robinson equation of state (WORK built-in). Refrigerant expansion occurs in Joule-Thompson valves and the refrigerant pressure drop across the heat exchangers is neglected. The temperature-enthalpy data of the natural gas are given in Table 1. 6.1. Base Case Solution. The application of the new methodology results in an optimal solution with a compression power of 33.39 MW. The composite curves for this solution are given in Figure 9. The optimal solution reported by Lee,4 on the other hand, had a compression power of 26.60 MW, which is 20.6% lower than the one found with the new methodology. However, it must be considered that because the minimum temperature difference was not fully enforced, Lee’s solution ended up with an effective ?Tmin of approximately 1.2 °C. Hence, for the comparison to be really meaningful, the base case was reoptimized with the new methodology and ?Tmin of 1.2 °C, ?nding an optimal solution that features 24.53 MW of power, that is, 7.8% lower than in Lee’s solution. The three solutions discussed are reported in Table 2. Note that a temperature approach as tight as 1.2 °C may result as impractical as it is unlikely to handle the normal operational variations that may occur in the plant once constructed. In practice it is unlikely that temperature approaches of less than 3 °C are considered during design. Solving the base case involved the optimization of seven variables and took 410 min in a Pentium IV processor (3.0 GHz) with 512 Mb of RAM. 6.2. Effect of Multistage Compression. It is worth noting that all three solutions presented in the previous section have a compressor outlet temperature of no less than 140 °C. A gas undergoing compression at such temperatures certainly occupies a relatively high speci?c volume and hence demands a relatively high speci?c compression power. To illustrate the effect of multistage compression, another design task was performed. This time the maximum stage pressure ratio was set to 5 and the minimum temperature difference back to 5 °C. The resulting
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8733
Figure 10. Compression train for optimal solution with maximum stage pressure ratio of 5.
Figure 11. ?Tmin sensitivity for two-staged refrigeration cycles.
minimum power solution has total compression power of 27.87 MW, which is 16.8% lower than the minimum power found when optimizing for a single compression stage (33.49 MW). Details are shown in Figure 10. The compression train features three stages with a pressure ratio of 2.95 each. The total intercooler duty was also reduced from 53.67 to 48.15 MW. Although the refrigerant composition is similar to that of the base case optimal solution, meaningful differences are observed in the refrigerant ?owrates (2.81 against 3.47 kmol/s before) and in the compressor inlet and outlet pressures (1.76 against 2.40 bar and 45.06 against 36.95 bar). It seems that now that the compression process is more ef?cient, the system can afford a larger total pressure ratio (25.6 against 15.4) in exchange for a reduction in the refrigerant ?owrate, the net effect being the observed reduction in total power while still keeping good care of temperature feasibility in the cryogenic exchanger. If the compression ratios are not assumed identical but allowed to vary as additional degrees of freedom during optimization, the optimal power exhibits a slight further reduction to 27.57 MW. In that case the compression ratios for the three compression stages are 3.57, 2.83, and 2.30, respectively, which are approximately within (20% of the 2.95 that resulted when assumed identical. One could be tempted to say that a multistage compression model is not needed at the synthesis stage because the power consumption can be reduced after optimizing the cycle for a single compression stage by just increasing the number of compression stages afterward. However, doing so in the case study reduces the total power only to 30.18 MW; still 2.31 MW above the 27.87 MW found when considering the effect of multiple stages during optimization. One possible reason for this is the unattractiveness of exploring high total pressure ratios when minimizing power in a cycle with a single compression stage, as discussed in the previous paragraph. 6.3. Two-Stage Refrigeration and Capital Cost Effect. A two-stage MR cycle was also optimized in a sensitivity analysis with a maximum stage pressure ratio of 5. The goal was to
Figure 12. Composite curves for the optimal MR system with four refrigeration stages.
picture the effect that the prede?ned minimum temperature difference has on the minimum power solutions. The effect on the total compression power can be seen in Figure 11a, where the resulting power of each solution is plotted against the respective value of ?Tmin. The monotonic trend was to be expected. The closer the hot and cold composite curves are allowed, the lesser the irreversibilities in the process and hence the lesser the total compression power. Note that the optimization for a ?Tmin of 5 °C gave a solution with a power consumption of 26.58 MW. This is 1.29 MW lower than the solution reported in the previous section and due to the fact that the refrigeration stages have been increased to two, which in this case is helping to achieve a better match between the composite curves in each stage. The capital cost of the solutions appearing in Figure 11a are plotted in Figure 11b also against the minimum temperature difference. The capital cost considered for any given solution includes the cost of the compressor and the cost of the cryogenic heat exchangers. The latter are estimated using the method in ESDU11 for aluminum-brazed plate-?n heat exchangers (PFHE), which estimates the volume and cost of these on the basis of the composite curves and typical heat transfer coef?cients given the type of application. This time the trend is not monotonic. A
8734 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 13. General cascade of two MR cycles.
This reduction in the number of compression stages, albeit there being the slightly increased total power, provided an additional contribution toward the capital minimization due to the economy of scale. 7. Refrigeration Systems in Cascade In the previous case study, using two stages of refrigeration allowed a 4.6% of reduction in the total compression power when compared to a single refrigeration stage system with multistage compression. In that case, a better match between the composite curves was achieved because a favorable change of the refrigerant ?owrate and average composition occurred at an intermediate temperature. However, this is not to be generalized for systems with more refrigeration stages. It is not always true that the more refrigeration stages there are in an MR system of the type in Figure 5a, the lower is the total compression power. In fact, when three refrigeration stages are used in the given example, the minimum power solution requires 26.95 MW of compression power, a slight increase over the 26.58 MW of the optimal two-stage system, although still less than the 27.87 MW of the optimal single-stage system. Furthermore, the lowest possible compression power resulted in 32.94 MW for an MR system with four refrigeration stages, which is considerably more than in the case of one-, two-, and three-stage systems. The above behavior is because, with each additional refrigeration stage, an intermediate separation of the vapor and liquid refrigerant streams is carried out, and this may introduce an unfavorable change in the refrigerant ?owrates and compositions in the downstream refrigeration stage(s). These changes in turn may make it dif?cult to achieve a good match between the composite curves because the designer does not have a complete control on the former. An illustration of this is shown in Figure 12, which corresponds to the optimal four-stage system. The refrigerant ?owrates and compositions in the different stages are not completely independent from each other. They are very much a consequence of the chosen overall refrigerant composition (the one through compression) and the intermediate temperature between stages. An alternative to increase the complexity within the same refrigeration cycle to achieve a more ef?cient overall system is to employ refrigeration systems in cascade. This is particularly
Figure 14. Optimization strategy for MR cycles in cascade.
minimum capital cost seems to exist somewhere between 1 and 5 °C. This behavior is also not surprising. As the total power decreases with decreasing ?Tmin, so does the compressor cost. However, ?Tmin has the reverse effect on the volume (and cost) of the PFHEs. The closer the composite curves are to each other, the smaller the temperature driving forces and hence the larger the heat transfer area, volume, and cost. With the purpose of further investigating this compressor/ PFHE capital tradeoff, the minimum temperature difference permitted during optimization was decreased to a negligible amount and the objective of the optimization was changed to minimum capital cost. In this way there would be no restrictions on the temperature differences, apart from avoiding crossovers, as long as the capital is minimized. The resulting solution is also shown in Figure 11. In the point of closest approach between hot and cold composite curves the temperature difference was 1.3 °C. The total compression power is located slightly above the power curve of minimum power solutions. However, the capital cost is considerably below the cost curve of minimum power solutions. The reason for this is that, apart from exploiting the compressor/PFHE capital tradeoff, the resulting pressure levels of the minimum capital solution are such that the compression task can be accommodated in just two stages, against the three needed in all of the minimum power solutions.
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8735
useful for refrigeration over wide temperature ranges and means sharing the refrigeration duty between two or more cycles, where the colder cycle(s) reject heat to the warmer one(s) and eventually the warmest cycle rejects heat to an ambient utility. The process stream(s) release heat to these cycles in decreasing order of temperature for more ef?ciency. This is releasing as much heat as possible to the warmest cycle, down to the ?rst so-called partition temperature, then releasing as much heat as possible to the second warmest cycle, down to the second partition temperature, and so on. The cycles within the cascade do not have to be of the same nature (e.g., pure refrigerant, MR, expander cycles). Cascaded cycles are commonly applied in the LNG industry. 7.1. Optimal Design of MR Cycles in Cascade. A general cascade of two MR cycles is shown in Figure 13. The process stream(s), with an inlet temperature TPIN, release heat in the upper (warmer) MR cycle, with NRU refrigeration stages, until reaching the partition temperature TP, and then undergo(es) further cooling in the lower (colder) MR cycle, with NRL refrigeration stages, until the required ?nal temperature TPOUT is reached. After compression, the refrigerant in the lower cycle rejects as much heat as possible to an external cold utility down to the temperature TRIN before further heat rejection in the NRU stages of the upper cycle, down to the partition temperature TP. Then it is routed to the ?rst of the NRL stages of the lower cycle, where it undergoes self-cooling, phase splitting, and expansion as in previously described noncascade MR systems. In the upper cycle, the upper refrigerant rejects heat only to the external cold utility down to the temperature TRIN before being sent to the ?rst of the NRU refrigeration stages. The upper refrigerant is to provide cooling to the process stream(s) and to the lower refrigerant. The approach previously described in detail in the optimal design of (noncascade) MR cycles can be adapted to a cascade MR cycles without much dif?culty. The basic idea of the optimization framework in Figure 8 is still valid in this case and an adaptation to a cascade cycle is presented in Figure 14. The optimizer proposes a series of candidate solutions to be evaluated by the simulator but this time the degrees of freedom in both the upper and the lower cycles, along with the partition temperature, are manipulated by the optimizer. The simulator is to apply basically the same problem formulation described previously to each cycle in order to evaluate the objective function. As the stage heat duties in the upper cycle depend on the ?owrate and properties of the MR stream received from the lower cycle, the latter is simulated ?rst. After this, the lower MR cooling curve from TRIN to TP is computed and combined with the process composite curve from TPIN to TP to produce the “effective” process composite curve in the upper cycle. This effective curve then de?nes the actual external refrigeration duties in the upper cycle. The calculation of this lower cycle cooling pro?le and of the effective process composite curve for the upper cycle is represented by eq 69-72, which must be added to the problem formulation in order to solve the upper cycle. The effective composite curve data TPREF and HPREF would replace TPR and HPR in the rest of the formulation when applied to the upper cycle.
OUT TLR ) TPROF(TIO, TP, PHOUT L 0, (PHL 0 - ?PLR), ZL1) (69) OUT HLR)HPROF(TI0, Tp, PHOUT L 0,(PHL 0 - ?PLR), ZL1, FL1 (70)
TPREF ) TCOMP(TPRU, TLR, HPRU, HLR) HPREF ) HCOMP(TPRU, TLR, HPRU, HLR)
(71) (72)
Other minor adjustments in the formulation include replacing OUT TPIN and TRIN with TP and PH0 with (PHOUT L 0 - ?PLR) in the lower cycle model, except for the equations related to compression, and TPOUT with TP in the upper cycle model. The objective function remains as a unique equation and has to be rede?ned as a function of the features in both the lower and the upper cycles. The number of stages for both cycles can be treated as integer variables, and consequently overall optimization framework can be formulated with mixed-integer nonlinear programming (MINLP). However, the optimization has been carried out with nonlinear programming (NLP) with prespecifying the number of stages employed in each cycle, in order to avoid computational dif?culties associated with the introduction of integer variables. 8. Case Study 2 - MR Cycles in Cascade The approach for the optimal design of MR cycles in cascade will be illustrated by tackling a problem previously published as a case study by Vaidyaraman and Maranas.6 A natural gas stream contains 93 mol % of methane, 5 mol % ethane, 1.5 mol % propane, and 0.5 mol % n-butane and is to be chilled from an inlet temperature of 19.85 to -58.15 °C at a constant pressure of 42 bar. A cascade of two MR cycles, with a varied number of refrigeration stages each, provides the required cooling. The hot liquid refrigerant streams are not subcooled in the refrigeration stages and hence the variations b of eqs 13, 21, 22, 24, 40 and 42 must be used (eqs 32, 33, 41,and 44are not required.). Contrary to the principles of an ef?cient system, the process stream is not precooled in the upper MR cycle but fed directly to the lower MR cycle. The former is only used for heat rejection from the latter. As the intention of this case study is to compare the results of the new design approach to those from Vaidyaraman and Maranas,6 this ?owsheet feature was left untouched in order to make the comparison on the same basis. Considering the above features, the generic ?owsheet used in this case study is shown in Figure 15. A further slight adjustment of the problem formulation is required to model such a process, the process stream(s) will not contribute to the effective process composite curve in the simulation of the upper cycle (i.e., TPRU and HPRU do not participate in eqs 71 and 72) and second, the process stream(s) enter the lower cycle at the temperature TPIN instead of the partition temperature TP. The refrigerants in the lower and the upper cycles contain ethane, propane and n-butane. After compression in a single stage, these reject heat to an external cold utility until reaching a temperature of 36.85 °C. The pressure drop of the refrigerants through the system is neglected and the minimum temperature difference allowed in the main heat exchangers is 2.5 °C. The Soave-Redlich-Kwong equation of state provides the basis for physical property calculations, which for this case study are performed by interfacing with Aspen Properties. The objective of the optimization is to minimize the coef?cient of performance of the system for different combinations in the number of lower and upper refrigeration stages. The coef?cient of performance (COP) in this case is de?ned as the ratio between the total compression work and the amount of heat released by the process stream. According to this de?nition, the lower the COP the more ef?cient the system. If an arbitrary ?owrate for the process stream is chosen, then minimizing COP would in practice be the same as minimizing total compression power as the amount of heat released by the process is ?xed.
8736 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Figure 15. Generic ?owsheet for case study 2. Table 3. Optimal COPs Obtained with the New Approach NRU 1 2 3 4 NRL ) 0 (no cascade) 0.5291 0.5238 0.5488 NRL ) 1 0.5415 0.4485 0.4266 0.4790 NRL ) 2 0.5024 0.4755 0.4263 0.4581 NRL ) 3 0.4566 0.4951 0.4805 -
Table 4. Optimal COPs obtained by Vaidyaraman and Maranas6 NRU 1 2 3 4 NRL ) 0 (no cascade) 0.5022 0.4562 0.4948 NRL ) 1 0.5890 0.4547 0.4144 0.4226 NRL ) 2 0.4608 0.4095 0.3957 0.4298 NRL ) 3 0.4623 0.4298 0.4124 -
Table 5. Relative COP with the New Approach (100 × COPNEW/COPPREV) NRU 1 2 3 4 NRL ) 0 (no cascade) 105.33 114.80 110.88 NRL ) 1 91.91 98.61 102.85 113.31 NRL ) 2 109.00 116.08 107.77 106.55 NRL ) 3 98.73 115.17 116.48 -
The pressures of the refrigerants at the compressor inlet are allowed to vary between 1.0 and 2.5 bar, the discharge pressures between 4 and 10 bar, the compression ratios between 1.5 and 12 (but only a single compression stage) and the refrigerant ?owrates between 0.1 and 1.0 kmol/s, which is adequate for an assumed 1 kmol/s of natural gas. The partition temperature is allowed to vary between -40 and 10 °C when a cascade is present. The optimal COPs encountered when applying the proposed approach are presented in Table 3 for different combinations of the number of stages in the lower cycle (NRL) and those in the upper cycle (NRU). Details on each of these solutions can be found in the Appendix. The COPs obtained with the previous approach6 are shown in Table 4 for comparison purposes. For more simplicity, Table 5 shows the relative improvements in the system performance (COP) achieved with the new approach. In only three cases it was possible to obtain lower COPs, with the best saving over the previous method being 8.09%, in the case of one stage in the lower cycle and one in the upper cycle. In most cases applying the new method resulted in higher COPs, by up to 16.48%. However, contrary to what one could initially think, this is evidence of the strength of the new method rather than a weakness, as all the new solutions
Figure 16. Composite curves of the previous solution for three stages in the lower cycle and three in the upper cycle.
are feasible ones. As discussed previously in the literature review, the method by Vaidyaraman and Maranas6 enforced temperature feasibility only at the ends of each refrigeration stage. As a result, temperature violations are plentiful in the previous results and an increase in the total power seems to be the only way to regain feasibility. For instance, assuming 1 kmol/s of natural gas, Figure 16 shows the composite curves of the lower and upper cycles that correspond to the previous optimal solution for a system with three stages in the lower cycle and three in the upper cycle. This is the same case in which the new method yields the highest relative COP increase as in Table 5. While the upper cycle exhibits only a slight ?Tmin violation at the cold end of the second stage, this violation is quite signi?cant in the second
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8737
?owrates in exchange for increased compression ratios. The net effect is, as previously mentioned, a 16.48% of increase in the COP (and total power) of the system but, nevertheless, a full compliance with temperature feasibility. The size of the problems solved in this case study ranged between 5 and 16 variables subject to optimization, each case taking between 77 and 295 min to solve in a Pentium IV processor (3.0 GHz) with 512 Mb of RAM. Typical GA parameters used in the optimization are population size ) 1000, number of generations ) 600, crossover probability ) 0.85, mutation method ) one-point mutation with adjustable rate based on ?tness, relative ?tness differential ) 0.5, steady-state-delete-worst reproduction, elitism technique is applied. 9. Conclusions The design of MR cycles is a challenging task. Also, local optima are abundant. This paper has described and illustrated the application of a new formulation and optimization strategy for the synthesis of MR cycles. It considers multistage refrigerant compression, full enforcement of the minimum temperature difference along the temperature pro?les, simultaneous optimization of variables, incorporation of capital costs in the objective function and the use of stochastic optimization (genetic algorithm) to overcome local optima. The effectiveness of the method was illustrated by ?nding improved and feasible solutions for two previously published case studies. The incorporation of multistage compression proved a key factor in the design of more ef?cient cycles since it not only reduces the power consumption per se, but also encourages the full exploration of total pressure ratios, unlike the single stage case. Although only compressor and PFHE capital costs were considered, the tradeoff between these was illustrated, and how in practice this means that the cost of power-related items is dominant in this type of systems. When applied to MR cycles in cascade, in some cases it was possible to obtain new designs with improved thermodynamic performance against the results published in previous work. In most cases, however, the new solutions did not apparently exhibit a performance improvement but nevertheless they were fully compliant with temperature feasibility checks, unlike the previous work that enforced this only at the ends of the heat exchangers and, as a consequence, intermediate ?Tmin violations and sometimes temperature crossovers occurred. Although it makes the solving process time-consuming and it still does not guarantee global optima, the application of a genetic algorithm optimizer to this design problem permits accepting the optimality of the results with greater con?dence than deterministic methods. Acknowledgment First author of this paper would like to thank the Centre for Process Integration at the University of Manchester and the Overseas Research Student Award Scheme for the ?nancial sponsorship of this project. Appendix Details on each of solutions found in case study 2 are listed in Tables A1-A4. Refrigerant ?owrates are per 1 kmol/s of natural gas.
Figure 17. Composite curves of the new solution for three stages in the lower cycle (a) and three in the upper cycle (b). Table 6. Comparison of Solutions for Three Stages in the Lower Cycle and Three in the Upper Cycle previous approach TP (°C) FL1 (kmol/s) PLIN L 0 (bar) PHOUT L 0 (bar) ZL 1 mol % C2H6 C3H8 n-C4H10 TI L (°C) FU 1 (kmol/s) IN PLU 0 (bar) OUT PHU 0 (bar) ZU 1 mol % C2H6 C3H8 n-C4H10 TI U (°C) 14.59 0.499 2.50 7.00 32.80 33.34 33.86 -11.14, -35.01 0.552 2.50 4.20 0.12 12.76 87.12 30.85, 16.53 new approach 0.32 0.396 2.36 7.02 46.62 29.58 23.80 -29.09, -51.17 0.4747 2.50 5.22 15.98 6.81 77.21 18.07, 3.44
stage of the upper cycle. Furthermore, a slight temperature crossover occurs. On the other hand, the composite curves of the optimal solution found with the new approach are those in Figure 17, showing no crossovers or minimum temperature approach violations. A more detailed comparison of both solutions is presented in Table 6, with meaningful differences found in most variables, contrary to the suggestion by Vaidyaraman and Maranas6 that any resulting infeasibility could be recti?ed with slight changes in pressure and still keep optimality. In general the new solution (1) allocates a larger share of refrigeration to the lower cycle than the previous solution (the partition temperature is now more than 14 °C lower), (2) features more methane and less propane and n-butane in the refrigerant mixtures, and (3) is able to afford a reduction in the refrigerant
8738 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008
Table A1. Solutions with One Refrigeration Stage in the Lower Cycle refrigeration stages in upper cycle variable TP (°C) FL1 (kmol/s) PLIN L 0 (bar) PHOUT L 0 (bar) ZL 1 mol % C2H6 C3H8 n-C4H10 TI L (°C) FU 1 (kmol/s) IN PLU 0 (bar) OUT PHU 0 (bar) ZU 1 mol % C2H6 C3H8 n-C4H10 TI U (°C) 0 (no cascade) 1 2 3 -21.78 -16.39 -44.69 0.2019 0.2187 0.1834 1.398 1.533 1.911 7.064 9.588 9.069 0.5069 0.1878 0.3053 0.2381 1.100 8.849 0.0085 0.5701 0.4214 0.5297 0.2843 0.1860 0.3414 2.109 6.421 0.0546 0.3353 0.6102 4.71 0.5649 0.0474 0.3877 0.3703 2.500 8.773 0.1913 0.2599 0.5488 3.03, -26.50 Table A4. Solutions with four refrigeration stages in the lower cycle refrigeration stages in upper cycle variable TP (°C) FL1 (kmol/s) PLIN L 0 (bar) PHOUT L 0 (bar) ZL 1 mol % C2H6 C3H8 n-C4H10 TI L (°C) FU 1 (kmol/s) IN PLU 0 (bar) OUT PHU 0 (bar) ZU 1 mol % C2H6 C3H8 n-C4H10 TI U (°C) 0 (no cascade) 0.6337 2.461 9.690 0.3820 0.1626 0.4554 10.60, -21.08, -50.51 -11.67 0.5926 2.499 4.397 0.3902 0.4723 0.1375 -24.20, -39.60, -50.86 0.3140 1.743 9.042 0.0000 0.6131 0.3869 1 11.55 0.5060 2.499 7.193 0.4154 0.2893 0.2953 -12.61, -33.05, -50.58 0.3239 2.451 4.845 0.0322 0.1570 0.8108 17.58 2 3 -
Nomenclature
Table A2. Solutions with Two Refrigeration Stages in the Lower Cycle refrigeration stages in upper cycle variable TP (°C) FL1 (kmol/s) PLIN L 0 (bar) PHOUT L 0 (bar) ZL 1 mol % C2H6 C3H8 n-C4H10 TI L (°C) FU 1 (kmol/s) IN PLU 0 (bar) OUT PHU 0 (bar) ZU 1 mol % C2H6 C3H8 n-C4H10 TI U (°C) 0 (no cascade) 0.4138 1.421 9.618 0.3636 0.0870 0.5494 -15.45 1 3.29 0.3445 2.261 7.980 0.4244 0.3255 0.2501 -29.52 0.2106 1.243 4.610 0.0216 0.0468 0.9315 2 -6.82 0.3529 1.675 4.623 0.3241 0.4566 0.2193 -37.63 0.3804 2.499 7.613 0.0067 0.5384 0.4549 3.60 3 -5.92 0.3209 2.494 8.310 0.5263 0.2639 0.2098 -36.24 0.5236 2.477 5.674 0.1759 0.0973 0.7268 18.73, 0.06
Subscripts i ) compression stage index L ) lower refrigeration cycle n ) refrigeration stage index U ) upper refrigeration cycle Parameters ?PCOLD ) total cold refrigerant pressure drop ?PHOT ) total hot refrigerant pressure drop ?PLR ) total pressure drop of the lower refrigerant though the upper cycle ?Tmin ) minimum temperature difference or temperature approach ηc ) isentropic ef?ciency of compressor ηLEX ) isentropic ef?ciency of liquid expander ηp ) isentropic ef?ciency of pump HPR ) set of enthalpy ?ows de?ning the overall process composite curve NR ) number of refrigeration stages PRMAX ) maximum pressure or compression ratio PRMIN ) minimum pressure or compression ratio TPIN ) process inlet temperature to the system TPOUT ) process outlet temperature from system TPR ) temperatures in the overall process composite curve TRIN ) temperature of hot streams after heat rejection to air or cooling water YLEX ) binary parameter indicating the use of liquid expanders (when set to 1) Functions CROP ) extracts a set of enthalpy ?ows or temperatures contained within a given temperature range from a larger array HCOMP ) combines stream data and returns the set of enthalpy ?ows de?ning a hot composite curve HEV ) returns the enthalpy ?ow that corresponds to a sample temperature in a composite curve HPROF ) returns a set of enthalpy ?ows de?ning the composite curve of a stream with the given ?owrate and compositions and within the given temperature and pressure range hsP ) returns the enthalpy of a stream given its composition, entropy and pressure hTP ) returns the enthalpy of a stream given its composition, temperature and pressure OBJ ) user-de?ned objective function. may change from case to case, including economic and/or performance criteria
Table A3. Solutions with Three Refrigeration Stages in the Lower Cycle refrigeration stages in upper cycle variable TP (°C) FL1 (kmol/s) PLIN L 0 (bar) PHOUT L 0 (bar) ZL 1 mol % C2H6 C3H8 n-C4H10 TI L (°C) FU 1 (kmol/s) IN PLU 0 (bar) OUT PHU 0 (bar) ZU 1 mol % C2H6 C3H8 n-C4H10 TI U (°C) 0 (no cascade) 0.5821 2.490 9.935 0.3797 0.1833 0.4369 3.26, -30.04 1 0.18 0.4430 2.479 5.800 0.3638 0.4399 0.1963 -20.87, -40.75 0.2548 2.500 8.601 0.0076 0.5412 0.4512 2 5.04 0.4504 2.428 6.140 0.3617 0.3942 0.2441 -19.40, -40.91 0.3493 2.469 5.519 0.0259 0.2717 0.7024 14.18 3 0.32 0.3959 2.363 7.024 0.4662 0.2958 0.2380 -29.09, -51.17 0.4747 2.500 5.224 0.1598 0.0681 0.7721 18.03, 3.44
Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8739 sTP ) returns the entropy of a stream given its composition, temperature and pressure TCOMP ) combines stream data and returns the temperatures in a hot composite curve TDEW ) returns the dew temperature of a stream given its composition and pressure TEV ) returns the temperature that corresponds to a sample enthalpy ?ow in a composite curve ThP ) returns the temperature of a stream given its composition, enthalpy and pressure TPROF ) returns a set of temperatures de?ning the composite curve of a stream with the given ?owrate and compositions and within the given temperature and pressure range VF ) returns the vapor fraction of a stream given its composition, temperature and pressur ZVAP ) returns the vapor composition of a partially vaporised stream given its total composition, temperature and pressure ZLIQ ) returns the liquid composition of a partially vaporised stream given its total composition, temperature and pressure Variables F ) total molar ?owrate of hot refrigerant entering a given refrigeration stage FC ) total molar ?owrate of cold refrigerant entering a given refrigeration stage FCMP ) molar ?owrate of refrigerant entering a given compression stage FL ) liquid molar ?owrate of hot refrigerant entering a given refrigeration stage FP ) liquid molar ?owrate of refrigerant after intercooler in a given compression stage FV ) vapor molar ?owrate of hot refrigerant entering a given refrigeration stage HC ) set of enthalpy ?ows de?ning the composite curve of the cold refrigerant in a given refrigeration stage hCIN ) cold refrigerant enthalpy at the inlet of a given refrigeration stage HCCOMP ) set of enthalpy ?ows de?ning the composite curve of the cold streams in a given refrigeration stage hCMP ) enthalpy of refrigerant after a given compression stage hCOUT ) cold refrigerant enthalpy at the outlet of a given refrigeration stage hEXPN ) enthalpy of refrigerant after expansion in valve or liquid turbine in a given refrigeration stage HHCOMP ) set of enthalpy ?ows de?ning the composite curve of the hot streams in a given refrigeration stage HHL ) set of enthalpy ?ows de?ning the composite curve of the liquid hot refrigerant stream in a given refrigeration stage HHV ) set of enthalpy ?ows de?ning the composite curve of the vapor hot refrigerant stream in a given refrigeration stage HLR ) set of enthalpies de?ning the composite curve of the lower refrigerant through the upper cycle hP ) enthalpy of refrigerant after pumping in a given compression stage HP ) set of enthalpy ?ows de?ning the composite curve of the process stream(s) in a given refrigeration stage HPREF ) set of enthalpy ?ows de?ning the effective process composite curve in the upper cycle hISCMP ) isentropic enthalpy of compression in a given compression stage hISLEX ) isentropic enthalpy of expansion of cold liquid refrigerant in a given refrigeration stage hISP ) isentropic enthalpy of pumping in a given compression stage NC ) number of compression stages ObjectiVe ) objective function value PLIN ) pressure of cold refrigerant at the inlet of a given refrigeration stage PHOUT ) pressure of hot refrigerant at the outlet of a given refrigeration stage PR ) pressure ratio of a given compression stage when dissimilar values are allowed. unique stage pressure ratio otherwise PCMP ) discharge pressure in a given compression stage QC ) heat received by the cold refrigerant in a given refrigeration stage QCMP ) heat removed in intercooler after a given compression stage QL ) heat removed from the hot liquid refrigerant in a given refrigeration stage QP ) heat removed from the process stream(s) in a given refrigeration stage QV ) heat removed from the hot vapor refrigerant in a given refrigeration stage TC ) set of temperatures de?ning the composite curve of the cold refrigerant in a given refrigeration stage TCCOMP ) set of temperatures de?ning the composite curve of the cold streams in a given refrigeration stage TCIN ) cold refrigerant temperature at the inlet of a given refrigeration stage TCOUT ) cold refrigerant temperature at the outlet of a given refrigeration stage TEXPN ) temperature of refrigerant after expansion in valve or liquid turbine in a given refrigeration stage th ) sample temperature for feasibility evaluation THCOMP ) set of temperatures de?ning the composite curve of the hot streams in a given refrigeration stage THL ) set of temperatures de?ning the composite curve of the liquid hot refrigerant stream in a given refrigeration stage THV ) set of temperatures de?ning the composite curve of the vapor hot refrigerant stream in a given refrigeration stage TI ) temperature of the hot streams after a given refrigeration stage (intermediate temperature) TLR ) set of temperatures de?ning the composite curve of the lower refrigerant through the upper cycle TP ) set of temperatures de?ning the composite curve of the process stream(s) in a given refrigeration stage TPREF ) set of temperatures de?ning the effective process composite curve in the upper cycle WLEX ) work produced in liquid turbine at a given refrigeration stage WC ) work required in a given compression stage WP ) pumping work required after a given compression stage X ) molar composition of hot liquid refrigerant in a given refrigeration stage XCMP ) molar composition of liquid after intercooler in a given compression stage Y ) molar composition of hot vapor refrigerant in a given refrigeration stage YCMP ) molar composition of vapor after intercooler in a given compression stage Z ) molar composition of cold refrigerant in a given refrigeration stage
Literature Cited
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ReceiVed for reView April 2, 2008 ReVised manuscript receiVed August 4, 2008 Accepted August 12, 2008 IE800515U