Optimal Design of Mixed Refrigerant Cycles

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 ofManchester, 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 mathematicalprogramming and offers significant improvements in relation to previous approaches. It includes multistagerefrigerant 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 refrigerantcycles or to systems consisting of two of these in cascade. The effectiveness of the method is illustrated byrevisiting previously published liquified natural gas case studies, for which better and feasible solutions areproduced, and which prove the importance of considering multistage compression and capital costs duringoptimization. The application of genetic algorithms in the design of mixed refrigerant cycles permits a greaterconfidence in the optimality of the results.

1. Introduction

Most low temperature processes feature one or more refrig-eration cycles with the purpose of removing heat from subam-bient hot streams. The provision of a cryogenic cooling requiressignificant power demands for compression, and it is veryimportant to achieve high energy efficiency in the design andoperation of refrigeration cycles, leading to low carbon emis-sions to the environment.

In a simple system with a closed refrigeration cycle, the heatis removed by vaporization of a low pressure refrigerant whichis then compressed and condensed at a higher pressure againsta warmer cold utility or heat sink. The condensed liquid is letdown in pressure (and temperature) by means of an expandingdevice such as a throttle valve. When cooling for a widetemperature range is required, a complex arrangement (forexample, a cascade cycle or a cycle with multilevel cooling) isintroduced to improve thermodynamic efficiency of the refrig-eration systems.

On the other hand, using mixed refrigerant (MR) in therefrigeration cycles provides very promising potential to yieldmore efficient, yet simple and reliable systems in comparisonto pure refrigerant ones, because a mixture of refrigerants isevaporated isobarically, not at a single but in a range oftemperatures. Although there are important “natural” applica-tions for MR cycles (e.g., LNG), their optimal design has notbeen the object of extensive research as in the case of purerefrigerant systems. Therefore, to explore advantages usingmixed refrigerants in low temperature cooling, it is the aim ofthis paper to develop a systematic design and synthesisoptimization framework for MR systems. The new designmethodology provides systematic investigation of design inter-actions as well as optimal and economic design of MR systems.

First, a brief overview is made regarding the design problemsbeing tackled in the area of mixed refrigeration systems, whichis followed by the review of existing literature as relevant tothe optimal design of MR systems, identifying improvementopportunities and gaps in the existing design practice. Next,the optimal design of MR systems is presented, in which the

details of the mathematical formulation and optimization strategyare provided for both single MR systems and MR systems incascade, including multistage compression and typical flowsheet/equipment variations. Finally, the proposed approach is il-lustrated with two case studies.

2. MR Systems

2.1. Pure Refrigerant and MR Cycles. A major limitationof vapor compression cycles that make use of pure componentsas refrigerants is that refrigeration is provided at a constanttemperature while the cold refrigerant is evaporating. For achosen refrigerant, the temperature at which cooling is providedis a consequence of the evaporator pressure, that is, the saturationtemperature.

If the hot streams (streams that need cooling) demand thecooling task to be carried out along a wide temperature range,a system providing all the refrigeration at a single level is likelyto have a poor performance. This is because large temperaturedifferences would exist in the heat exchanger(s), moving thesystem away from thermodynamic reversibility, and hence, fromthermodynamic efficiency. That being the case, a multilevel purerefrigerant system is likely to be implemented, in an attempt toreduce the temperature differences in the system, as seen inFigure 1, in which three pressure levels are used to providerefrigeration at three temperature levels. However, both heattransfer area and complexity would increase as a consequence.The refrigerant compressor would need as many stages asrefrigeration levels.

Multicomponent, or mixed refrigerants, unlike pure refriger-ants, undergo isobaric phase change through a range oftemperatures contained within the dew and bubble temperaturesof the mixture. Given the right pressures and compositions, agood match between the process and refrigerant temperatureprofiles can be obtained with a simple configuration, as shownin Figure 2. Although refrigerant compression can take placein several compression stages with intercooling, only one stageis shown in the illustration for simplicity. A practical maximumpressure ratio per compression stage (usually 4-5 for industrialapplications) is likely to define the number of compressionstages, rather than the number of pressure levels, since in thiscase the cold refrigerant evaporates at a unique pressure (orthrough a small pressure range, if friction pressure drop is

* To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +44 (0)161 306 8755. Fax: +44 (0)161 2367439.

Ind. Eng. Chem. Res. 2008, 47, 8724–87408724

10.1021/ie800515u CCC: $40.75 2008 American Chemical SocietyPublished on Web 10/18/2008

accounted for in the evaporator). Care should be taken, however,since the reduced average temperature difference would leadto increased heat transfer area. A minimum temperature differ-ence (∆T min) should be established for practical design purposes.Since the benefits of using MR cycles are highlighted insituations where refrigeration is to be provided along widetemperature ranges, these types of cycle find a natural applica-tion in the liquified natural gas (LNG) industry, where refrigera-tion is required from ambient temperature to around -160 °C.

2.2. Flowsheet Variations of MR Cycles. An interestingflowsheet variation in MR cycles is the one shown in Figure 3,where the high pressure refrigerant rejects heat not only to an

external heat sink but also to itself, once expanded. In this waythe hot refrigerant is further cooled down, probably subcooled,

Figure 1. A three-level pure refrigerant cycle.

Figure 2. A MR cycle.

Figure 3. A self-cooling MR cycle.

Figure 4. A three-stage MR cycle.

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8725

before expanding, allowing the cold refrigerant to reach a lowertemperature and/or a lower vapor fraction after the expansion.Such a benefit is achieved at the expense of a higher heat transferarea, since the total duty in the heat exchanger increasessignificantly. A multistream heat exchanger can be used tohandle all hot streams and the cold stream in a single piece of

equipment. This process is known as the Pritchard cycle and isdescribed in more detail by Walsh.1

Repeated partial condensation and separation of the refrigerantstream has been reported to achieve a better match between thetemperature profiles.2 However, this should not be taken as ageneral rule. Figure 4 shows a MR cycle with three refrigeration

Figure 5. MR structure used by Vaidyaraman and Maranas.6

Figure 6. MR system diagram.

8726 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

stages, as an extension of Figure 3a. Although increasing thenumber of stages may reduce the power consumption, the designalso grows in complexity and probably in capital cost. Thisresults in an important tradeoff for designers.

2.3. Multistage Compression. As compressors used inindustry have a practical maximum stage pressure ratio ofaround 4-5, it is very common to find compression tasksperformed in multiple stages. Further, if there is an appropriatecold utility, the partially compressed gas originating from a givenstage may be cooled down before entering the next compressionstage (intercooling). The lower temperature of the partiallycompressed gas reduces the volumetric flowrate and in conse-quence reduces the compression power of the next stage. If partof the gas condenses, then it is necessary to remove the liquidso that a dry vapor stream enters the next compression stage.The liquid removed is then pumped and mixed with the fullycompressed gas and with any liquid streams from otherintercoolers at the outlet of the last compression stage at samepressure. Condensate removal and pumping also helps in savingpower, as increasing the pressure of liquids is much cheaper inpower (and capital) than doing so with gases.

3. Literature Review

Although there are many publications on the analysis of MRsystems, only a few exist on their optimal design usingmathematical programming. The earliest attempt to optimizethese systems in a systematic manner was carried out by Ait-Ali.3 This work tackled the optimization of an MR system withthe same configuration as a two-level pure refrigerant system.The focus was largely on minimizing total compression powerby trying to enforce a constant temperature difference throughthe cryogenic heat exchangers. Although rich in practicalconsiderations and insights relevant to LNG production, refrig-erant pressures and flowrates were not the result of optimization,but rather set heuristically. Different solution procedures wererequired depending on the number of components present inthe refrigerant. Also, to handle the complexity of the problem,only binary and tertiary mixtures were considered and thethermodynamic accuracy had to be sacrificed by using anequation of state based on ideal solution assumptions andRaoult’s law for vapor-liquid equilibrium, which restrict thevalidity of the numerical results to low pressure and warmtemperature conditions. The optimization method was limitedto a two-dimensional numerical search. Refrigerant subcoolingand separation of liquid and vapor refrigerant streams at

intermediate temperatures were not addressed, neither wascascading 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. Givena refrigeration task in the form of a hot composite curve (acombined temperature-enthalpy profile of all hot streams) andthe number of refrigeration stages, his approach allows for theoptimization of key variables in a process flowsheet of the typeshown in Figure 4. In principle, the optimization variables arethe refrigerant composition and flowrate (at compressor inlet),the compressor inlet and outlet pressures, and the intermediatetemperatures (at which refrigerant is separated into vapor andliquid streams). However, the decision variables are notoptimized simultaneously. Instead, the refrigerant compositionsare optimized at fixed refrigerant flowrate and pressures usingnonlinear programming (NLP). Once the optimal compositionis obtained, the hot and cold temperature profiles are checkedfor feasibility. If they do not cross, then new refrigerant flowrateand pressures are proposed on the basis of heuristics, judgment,or optimization. And the procedure is repeated until no furtherimprovementispossiblewithoutincurringtemperatureinfeasibilities.

Lee4 also used three different types of objective function.Two of them tried to match the temperature profile of the coldrefrigerant to an ideal profile (the hot composite curve shifteddown by ∆Tmin) by minimizing either the maximum violationof the minimum temperature difference or the sum of suchviolations along the profile. The third possible objective functionwas the minimization of compressor power, but its use was notrecommended until the final part of the overall optimization.That is because, since there is no constraint on the vapor fractionat the inlet of the compressor, it is less likely to have wetnessat the compressor inlet toward the end of the overall optimizationdue to a probable lower value of the refrigerant flowrate.

The design of MR cycles is a highly nonlinear problem withmany local optima. One of the main drawbacks of Lee’sapproach is that the nonsimultaneous optimization of variablescoupled with the dependence of the final solution on the initialguess, on the nonsystematic selection of refrigerant flowrate andpressures (especially if updated using heuristics or judgment)and on the switching of the objective function, makes reachingthe global optimal solution (or a good near-optimal solution,for practical purposes) very unlikely.

In Lee’s method, the minimum temperature differencebetween the hot and cold temperature profiles is not fullyenforced, which often leads to very tight profiles, apart fromrelying on human intervention at each iteration to check for

Figure 7. Temperature feasibility check.


temperature feasibility. It is assumed that the refrigerantcompression takes place in one stage with no intercooling,regardless of the compression ratio. No capital costs areconsidered during the optimization procedure.

Lee4 also explored the synthesis of more complex systemssuch as MR cycles in cascade with pure refrigerant cycles. Inhis approach, although the partition temperature between thetwo cycles was subject to optimization, each cycle wasoptimized separately to minimize compression power for anassumed value of partition temperature. The partition temper-ature was then iterated manually in an outer loop until the totalpower was considered optimal. Again, this nonsimultaneous,nonsystematic optimization of variables is likely to lead tononoptimal solutions, as discussed before.

Another relevant precedent in the synthesis of multistageMR cycles is that by Vaidyaraman and Maranas,6 which isalso based on NLP. The cycle topology used by them is theone shown in Figure 5a. Although in principle it may lookvery different, it is just a slight variation of the type ofstructure in Figure 4, only that the liquid refrigerant streamis not subcooled after each flash step, as shown in Figure 5bas an alternative representation. The variables being opti-mized in this case were the refrigerant composition (at thestream being expanded in the last stage), the compressor inletand outlet pressures and the vapor fraction at flash drums 2to N. All the design variables (each with a fixed number ofstages) were optimized simultaneously.

Two key assumptions were made by Vaidyaraman andMaranas.6 The first one was that the hot refrigerant leaving stageN was at its bubble point, and the second one was that the coldrefrigerant streams leaving each stage were at their dew point.These assumptions, although not unreasonable, constrain thesolution space unnecessarily and could lead to good opportuni-ties missed by the optimizer. Another major shortcoming of themethod proposed is that temperature feasibility is only enforcedat the ends of the heat exchangers. This opens the possibilityof not only minimum temperature difference violations but alsoof temperature crossovers being overlooked during the optimi-zation. Although the authors suggest that, if violated, feasibilitymight be regained after the optimization by correcting therefrigerant pressures, doing so could lead the solution to loseits optimality.

As in Lee,4 Vaidyaraman and Maranas6 assumed single stagecompression without intercooling and did not consider capitalcosts. However, they made a further effort in trying to overcomelocal optima by performing a number of optimizations withdifferent starting points. The effect of refrigerant pressure dropswas not covered in their formulation.

In overall, the review of previous work on MR systemsreveals a quite significant potential for improvement. An idealdesign approach would be one that combines (a) flexibilityto handle flowsheet options such as multiple refrigerationstages, refrigerant subcooling, and cascade systems, (b)energy-efficient features, such as multistage compression, (c)a systematic optimization framework powerful enough toexploit the relevant degrees of freedom and to overcome thechallenges of a highly nonlinear problem, (d) no assumptionson the thermodynamic state of refrigerant streams just forthe sake of a simplified calculation procedure, (e) fullfeasibility enforcement, and (f) the cost consequences of thedesign decisions. A design approach like this is not yetavailable in the open literature, and in order to overcomeshortcomings addressed in above, a novel, systematic and

robust procedure for the optimal design of low temperatureprocesses is proposed in the next section.

4. Design and Optimization Frameworks

Figure 6 shows a generic multistage MR cycle and amultistage compression system with intercooling, indicatingsome of the nomenclature used in the formulation. Variablesfor the hypothetical refrigeration stage 0 and compression stage0 were included in order to simplify the formulation. Also, forthe sake of a compact formulation, some calculations are notdescribed explicitly. These are represented as functions of thetype f(x1,..., xN) and are mainly routine physical propertycalculations (e.g., enthalpy for a given set of compositions,temperature, and pressure) and temperature profile operations.It is assumed that all hot streams leaving each multistream heatexchanger are at the same temperature.

Objective Function. Equation 1 represents the objectivefunction as a generic function of the main process variables.This objective function may well change from case to caseaccording to the purpose of the designer (e.g., minimumcompressor power, minimum capital investment, minimum totalcost, etc.) and to the economic models used.

OBJECTIVE)OBJ(WC, QCMP, THCOMP, HHCOMP, TCCOMP, HCCOMP, ...) (1)

Stage Material and Energy Balances. Equations 2-7establish the relationship between the total flows of the streamsaround the system. Equations 2-4 describe the continuity ofvapor and liquid hot refrigerant stream flows from stage to stage.The hot vapor refrigerant stream is made unavailable at the NRth

stage because of the inexistence of a flash drum at stage NR-1(eq 5). All the material arriving to stage NR from stage NR -1is put through the hot refrigerant liquid stream for formulationpurposes, 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 (2)

Fn ) FVn-1 (3)

FVn )VF(Yn-1, TIn-1, PHn-1OUT) neNR- 1 (4)

FVNR ) 0 (5)

FCn ) FLn + FCn+1 neNR- 1 (6)

FCNR )FNR (7)

Vapor and liquid compositions are obtained from phaseequilibrium calculations at each flash drum (eq 8 and 9) althoughin practice the vapor and liquid compositions are obtainedsimultaneously from a single flash subroutine. The compositionof the only hot refrigerant stream in stage NR must be the sameas that of the hot vapor refrigerant at the previous stage (eq10). Equation 11 states that the total composition of the coldrefrigerant at each stage is the same as that of the hot vaporrefrigerant at the previous stage and is derived from componentbalances carried out around the first n - 1 stages as a group.

Yn )ZVAP(Yn-1, TIn-1, PHn-1OUT) (8)

Xn )ZLIQ(Yn-1, TIn-1, PHn-1OUT) neNR- 1 (9)

XNR ) YNR-1 (10)

Zn ) Yn-1 (11)

The heat removed from hot refrigerant streams at each stageis calculated in eq 12 and 13. The initial and final values of the


vector of intermediate temperatures (TI) are set to TRIN andTPOUT, respectively, for consistency with the input data (eqs14 and 15), although TI0 and TINR are not actual intermediatetemperatures. Since the refrigeration task is assumed to be givenin the form of a precalculated process composite curve (PR,HPR), eq 16 calculates the heat removed from the processstream(s) at each stage according to the enthalpy difference ofsuch a curve between the respective pair of adjacent intermediatetemperatures, the exception being stage 1 (eq 17), because inprinciple the initial process temperature (TPIN) might be differentfrom the initial hot refrigerant temperature (TI0 ) TRIN).Equation 18 ensures that the total heat removed from the hotstreams is absorbed by the cold refrigerant stream at each stage.As a result, the cold refrigerant outlet enthalpies and temper-atures can be determined as a result of an energy balance (eqs19 and 20).

QVn ) FVn[hTP(Yn, TIn, PHnOUT)- hTP(Yn, TIn-1, PHn-1

OUT)](12)

QLn ) FLn[hTP(Xn, TIn, PHnOUT)- hTP(Xn, TIn-1, PHn-1

OUT)](13)

TI0 )TRIN (14)

TINR )TPOUT (15)

QPn )EVAL(TPR, HPR, TIn)-

EVAL(TPR, HPR, TIn-1) ng 2 (16)

QP1 )EVAL(TPR, HPR, TI1)-EVAL(TPR, HPR, TPIN)

(17)

QCn ) - (QVn +QLn +QPn) (18)

hCnOUT )

FCnhTP(Zn, TCn,INPLn

IN)+QCn

FCn(19)

TCnOUT )ThP(hCn

OUT, PLn-1IN ) (20)

The enthalpy of the refrigerant after expansion is calculatedin eq 21. Depending on the value of the user defined binaryparameter YLEX, such an enthalpy will be set equal to eitherthe inlet enthalpy (for YLEX ) 0, corresponding to Joule-Thompson valves) or to a function of the isentropic enthalpyof expansion and a constant isentropic efficiency (for YLEX )1, corresponding to liquid expanders). The enthalpy of isentropicexpansion is calculated in eq 22. The refrigerant temperatureafter expansion is given in eq 23. When expanders are used, eq24 will calculate the associated power produced. Otherwise thispower would be implicitly forced to zero as in such case thereis no enthalpy difference.

hnEXPN ) (1-YLEX)hTP(Xn, TIn, PHn

OUT)+YLEX(1- ηLEX)hTP(Xn, TIn, PHn

OUT)+ ηLEXhISnLEX (21)

hISnLEX ) hsP(Xn, sTP(Xn, TIn, PHn

OUT), PLnIN) (22)

TnEXPN )ThP(Xn, hn

EXPN, PLnIN) (23)

WLEXn ) FLn(hnEXPN - hTP(Xn, TIn, PHn

OUT)) (24)

The inlet cold refrigerant temperature to any given stagefeaturing an inlet mixer (stages 1 to NR -1) can be calculated(Equation 26) after its enthalpy is known through an energybalance around the mixer (Equation 25). Since there is no mixer

at the inlet of stage NR, the inlet cold refrigerant temperatureis the same as the one after expansion (eq 27).

hCnIN )

FLn · hTP(Xn, TnEXPN, PLn

IN)+ FCn+1hTP(Zn+1, TCn+1OUT, PLn

IN)FCn

×

neNR- 1 (25)

TCnIN ) ThP(Zn, hCn

IN, PLnIN) neNR- 1 (26)

TCNRIN ) TNR

EXPN (27)

Pressure, Temperature, And Enthalpy Profiles. Accuratepressure drop predictions across heat exchangers, especially of thecompact type, may result in a quite lengthy calculation task, sincedetails of the actual exchanger geometry and internals (e.g., typeof fins), as well as a set of additional fluid transport properties, arerequired for such a purpose.7 If a rigorous calculation of such kindwas implemented in the present formulation it would increase themathematical complexity considerably, taking the focus away formthe main design variables. On the other hand, neglecting thepressure drops in the cycle could lead to overlooked infeasibilitiesbecause of an inaccurate prediction of temperature profiles and/orto a nonoptimal and/or subdesigned system. In the present for-mulation, an intermediate point is adopted. The total hot refrigerantpressure drop (PHNR

OUT -PH0OUT) has a fixed value defined by the

user before the optimization (∆PHOT), which allows the estimationof intermediate hot refrigerant pressures by approximating themas a linear function of the respective temperature, as shown in eq28. A similar approximation is performed to obtain the intermediatecold refrigerant pressures (eq 29). However, these are assumed asa linear function of the temperature of the hot refrigerant, insteadof that of the cold refrigerant for convenience in the sequentialsimulation approach.

PHnOUT ) PH0

OUT -TIn -TRIN

TPOUT -TRIN∆PHOT (28)

PLnIN ) PL0

IN +TIn -TRIN

TPOUT -TRIN∆PCOLD (29)

For a given composition and inlet and outlet temperatures, thetemperature-enthalpy profiles at each refrigeration stage are ob-tained by evaluating the stream enthalpy repeatedly at a numberof intermediate points, previously defined by the designer (i.e., thenumber of points, not the points themselves), within the respectivetemperature range. A linear behavior of refrigerant pressure withtemperature is also assumed during the calculation of thetemperature-enthalpy profiles. Cold refrigerant pressures, this time,are estimated according to the cold refrigerant temperatures andnot those of the hot refrigerant as done previously. Extra evaluationpoints are added at the exact dew and/or bubble points of themixture if found within the temperature and pressure range. Thefunctions TPROF and HPROF are in practice one single functionbeing shown separately for formulation purposes. The hot vaporrefrigerant, hot liquid refrigerant, and cold refrigerant profiles aredefined in eq 30 to 35.

THVn )TPROF(TIn-1, TIn, PHn-1OUT, PHn

OUT, Yn)neNR- 1 (30)

HHVn )HPROF(TIn-1, TIn, PHn-1OUT, PHn

OUT, Yn, FVn)neNR- 1 (31)

THLn )TPROF(TIn-1, TIn, PHn-1OUT, PHn

OUT, Xn) (32)


HHLn )HPROF(TIn-1, TIn, PHn-1OUT, PHn

OUT, Xn, FLn) (33)

TCn )TPROF(TCnIN, TCn

OUT, PLnIN, PLn-1

IN , Zn) (34)

HCn )HPROF(TCnIN, TCn

OUT, PLnIN, PLn-1

IN , Zn, FCn) (35)

Since the overall process composite curve is given as an inputto the problem, the process profiles at each stage are obtainedby just extracting the corresponding data within each respectiverange of temperatures. The overall process profile input to theproblem should consider the effect of pressure drop. The processdata extraction is performed in eq 36-39. If the processcomposite curve was not an input to the problem, the processprofiles could be constructed using a similar method as for therefrigerant profiles.

TPn )CROP(TPR, TIn-1, TIn) ng 2 (36)

TP1 )CROP(TPR, TPIN, TI1) (37)

HPn )CROP(HPR, TIn-1, TIn) ng 2 (38)

HP1 )CROP(HPR, TPIN, TI1) (39)

Equations 40-43 combine all the hot profiles to form a singlehot composite curve in each stage. Equations 44 and 45 obtainthe stage cold refrigerant composite curves.

THnCOMP )TCOMP(THVn, THLn, TPn, HHVn, HHLn, HPn)

neNR- 1 (40)

THCOMP

NR)TCOMP(THLNR, TPNR, HHLNR, HPNR) (41)

HHnCOMP )

HCOMP(THVn, THLn, TPn, HHVn, HHLn, HPn) ne

NR- 1(42)

HHCOMP

NR)HCOMP(THLNRTPNR, HHLNR, HPNR) (43)

TCnCOMP )TCOMP(TCn, HCn) (44)

HCnCOMP )HCOMP(TCn, HCn) (45)

Once the hot and cold composite curves are known at eachstage, the temperature feasibility at each stage can be evaluatedby comparing the temperature of the hot composite curve tothat of the cold one for a given enthalpy. Such comparison isperformed at each enthalpy point belonging to the hot compositecurve. If the temperature of the cold composite evaluated at agiven enthalpy is greater than that of the hot composite then atemperature cross occurs. If the temperature of the coldcomposite is lower than that of the hot composite by less than∆Tmin then a violation of the minimum temperature differenceoccurs. If the temperature of the cold composite is lower thanthat of the hot composite by at least ∆Tmin then the point isperfectly feasible. This temperature feasibility check is per-formed by eq 46 and further illustrated in Figure 7. Checkingfeasibility for a couple of points only will not ensure overallfeasibility for heat transfer throughout overall temperature range,whereas employing a large number of discrete points with verysmall interval will significantly increase computational time inthe optimization. Various numbers of points have been tested,and in this study, around 30 points (together with extraevaluation points, depending on the characteristics of compositecurves) were taken, which was enough to check feasibility ofheat recovery in exchangers without using excessive computa-tional resources.

th-TEV(HCnCOMP, TCn

COMP, HEV(THnCOMP, HHn

COMP, th))g∆Tmin ∀ th ∈ THn

COMP (46)

Compression. In practice, stages belonging to the samecompressor feature pressure ratios not too different from eachother. Identical pressure ratios are therefore not an unreasonableassumption. However, allowing dissimilar pressure ratios wouldmean more degrees of freedom during optimization, which maybring the advantage of an even further improvement of theobjective function value (e.g., total compression power) and/ora more flexible matching with mechanical drivers. The formula-tion in this work is flexible and allows both approaches;dissimilar compression ratios subject to optimization when fullexploitation of the degrees of freedom is required and anidentical compression ratio approach in case a simpler optimiza-tion is desired.

Equation 47 ensures that the values of stage pressure ratiosare consistent with the compressor inlet and outlet pressures.The discharge pressures of each compression stage are thencalculated by eq 48. Equations 49 and 50 enforce minimumand maximum limits to the stage pressure ratios. In case thedesigner would like to perform the optimization assumingidentical pressure ratios, the unique variable PR would replacethe set of variables PRi in the problem formulation. Equation51 states that the number of stages is such that thecompression task would not be possible with one stage less.Although this may appear redundant with eq 47, it isnecessary for the identical pressure ratio approach since inthat case the unique variable PR is not controlled by theoptimizer. As a consequence, there may be more that oneset of NC and PR values that satisfy eqs 47, 49, and 50simultaneously, and to ensure the number of stages is notexcessive, eq 51 has to exist in order to define the minimumnumber of stages as the design criteria in that situation.

PL0IN∏

j)1

NC

PRj ) PH0OUT (47)

PiCMP ) PL0

IN∏j)1

i

PRj (48)

PRie PRMAX (49)

PRig PRMIN (50)

PH0OUT > PL0

IN ∏j)1

NC-1

PRj (51)

Equation 52 avoids any wetness in the cold refrigerant streamfrom refrigeration stage 1, which is the stream entering thecompression train. Equations 53 and 54 make the compressionnomenclature consistent with that of the remainder of thecycle.

TC1OUTgTDEW(Y0, PL0

IN) (52)

P0CMP ) PL0

IN (53)

Y0CMP ) Y0 (54)

Equations 55-59 establish refrigerant flowrates and composi-tions around the compression system in consistency withphase equilibrium and stage material balances. It is assumedthat all the partially compressed streams can be cooled downto a temperature TRIN by an external heat sink and that anyliquid that is formed is separated, pumped to the final


pressure, and remixed before the final cooler, in accordancewith Figure 6b.

YiCMP )ZVAP(Yi-1

CMP, TRIN, PiCMP) ieNC- 1 (55)

XiCMP )ZVAP(Yi-1

CMP, TRIN, PiCMP) (56)

FiCMP )Fi-1

CMPVF(Yi-2CMP, TRIN, Pi-1

CMP) ng 2 (57)

F1CMP )F1 (58)

FiP ) Fi

CMP - Fi+1CMP ieNC- 1 (59)

Stage compression power is calculated in eqs 60 and 61on thebasis of the stage flowrate and enthalpy difference. Stage outletenthalpy is calculated using a constant isentropic efficiency ineq 62 and 63, the last one being derived from the definition ofisentropic efficiency. Stage pumping power is calculated in asimilar way by eqs 64-66

WCi )FiCMP(hi

CMP - hTP(Yi-1CMP, TRIN, Pi-1

CMP) ng 2

(60)

WC1 )F1CMP · (h1

CMP - hTP(Y0CMP, TC1

OUT, P0CMP)) (61)

hISiCMP ) hSP(Yi-1

CMP, sTP(Yi-1CMP, TRIN, Pi-1

CMP), PiCMP) (62)

hiCMP ) (1- ηc)hTP(Yi-1

CMP, TRIN, Pi-1CMP)+ ηchISi

CMP (63)

WPi ) FPi(hiP - hTP(Xi

CMP, TRIN, PiCMP)) ieNC- 1

(64)

hiP ) (1- ηp)hTP(Xi

CMP, TRIN, PiCMP)+ ηphISi

P

ieNC- 1 (65)

hISiP ) hsP(Xi

CMP, sTP(XiCMP, TRIN, Pi

CMP), PH0OUT)

ieNC- 1 (66)

Finally, eq 67 calculates the heat removed at each intercoolerexcept for the one after the last compression stage, which iscalculated in eq 68 based on an energy balance around the lastintercooler and all the mixers in conjunction.

QiCMP )Fi

CMP · (hTP(Yi-1CMP, TRIN, Pi

CMP)- hiCMP)

ieNC- 1 (67)

QNCCMP )F1hTP(Zi

CMP, TRIN, PH)-

∑ i)1

NC-1(hi

P ·FiP)- FNC

CMPhNCCMP (68)

Adaptation for Hot Liquid Refrigerant without Subco-oling. MR cycles without hot liquid refrigerant subcooling, asin Figure 5, are a particular case in the formulation above. Ifsuch a cycle was to be represented, some constraints wouldchange: Namely, all QLs would be equal to zero; the refrigerantexpansions would start at conditions TIn-1 and PHn-1

OUT, insteadof TIn and PHn

OUT; there would not be need for calculating anyhot liquid refrigerant profile (i.e., eqs 32, 33, 41, and 43). Theequations to be modified are shown below as variations b ofthe original equations:

QLn ) 0 (13b)

hnEXPN ) (1-YLEX)hTP(Xn, TIn-1, PHn-1

OUT)+YLEX(1- ηLEX)hTP(Xn, TIn-1, PHn-1

OUT)+ ηLEXhISnLEX (21b)

hISnLEX ) hsP(Xn, sTP(Xn, TIn-1, PHn-1

OUT), PLnIN) (22b)

WLEXn ) FLn(hnEXPN - hTP(Xn, TIn-1, PHn-1

OUT)) (24b)

THnCOMP )TCOMP(THVn, TPn, HHVn, HPn)

neNR- 1 (40b)

HHnCOMP )HCOMP(THVn, TPn, HHVn, HPn)

neNR- 1 (42b)

5. Solution Strategy

The optimization problem, as formulated in the previoussection, is of the NLP type: Minimize eq 1, subject to constraints2-64 (with variations 13b, 21b, 29b, 30b, and 37b-40b, ifappropriate).

However, not all the formulation has been explicit (e.g., phaseequilibrium and physical property calculations), and in practicepart of the calculations may take place in external subroutines(e.g., interfacing with commercial process simulators). Also, asstated before, the problem is nonlinear and features many localoptima. Traditional deterministic optimization methods wouldget readily trapped in these. On the other hand, stochasticmethods (e.g., genetic algorithm, simulated annealing) may offermore confidence on the optimality of the final solution at theexpense of computational time. A genetic algorithm (GA) hasbeen chosen as the main optimizer in this work. Geneticalgorithms try to copy nature regarding the evolution of species.They perform iterations not on a single candidate solution buton a population of candidate solutions (individuals) instead. Thepath toward optimality is built by sharing information betweenindividuals (reproduction) and by evaluating and ranking theresulting offspring according to their relative optimality (fitness)in a given number of major iterations (generations). Theinformation defining 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 advantageof GA is that no initial guess is needed to start up theoptimization process. Further details on how genetic algorithmswork can be found in books such as Goldberg8 and Stender etal.9

Pikaia,10 a genetic algorithm by the U.S. National Center forAtmospheric Research, has been adopted in this work andmodified in order to improve the quality of the initial generation.Randomly generated individuals are now evaluated before beingallowed into the initial generation. They qualify as a memberof the initial generation only if excessive infeasibilities are notincurred (e.g., average crossovers of no more than 3 °C and acompressor inlet temperature at most 2 °C below dew point),otherwise they are rejected. This filter, although it increases thecomputational time for the first generation considerably, ensuresa departure from a better set of candidate solutions, shorteningthe path to optimality. For case studies in this paper, 90∼95%of candidates are rejected during the filtration step.

Figure 8 illustrates the flow of information within the generaloptimization framework. The optimizer (GA in this case)proposes a series of candidate solutions and relies on thesimulator for their assessment. How the optimization taskevolves is decided by the optimizer on the basis of theassessment of the candidate solutions. The interactions betweenthe GA and the simulator result in a set of the best solutionsfound over a discretized solution space. Standard NLP optimiza-tion(s) can be carried out afterward having the best discretizedsolution(s) as initial guess in order to fine-tune and finally reportthe optimal solution on the basis of a continuous solution space.Both the optimizer and the simulator have been implemented


in FORTRAN 77 and within WORK, part of the process designsoftware suite available from the Centre for Process Integrationat the University of Manchester. This gives the user the choiceof using either built-in thermodynamic property models ordelegating these calculations to commercial process simulatorssuch as HYSYS or Aspen Properties that would run in thebackground.

The chosen independent variables for this application are therefrigerant flowrate (F1) and composition (Z1), the compressorinlet and outlet pressures (PH0

OUT and PL0IN) and the intermediate

temperatures (TI1 to TINR -1) for cycles with two stages or more.Given the values for all these variables, the simulator isresponsible for calculating the remaining variables and themagnitude of infeasibilities (e.g., temperature profiles andcompressor inlet wetness) and evaluating the objective function.Penalties may be applied to the objective function according tothe magnitude of the infeasibilities as a disincentive in theranking of solutions incurring violations. However, duringoptimization, care should be taken in removing design candidateswith considerable magnitude of infeasibilities in the objectivefunction since infeasible solutions (especially the slightly andmoderately infeasible) may still contain features worthy toinherit and evolve by offspring candidate solutions.

6. Case Study 1

The problem formulation and optimization strategy discussedin the previous section are put into practice using the same LNGcase study published by Lee.4 A pretreated natural gas streamis to be cooled from 25 °C down to -163 °C using a mixtureof nitrogen, methane, ethane, propane, and n-butane as refriger-ant in a single stage cycle (as in Figure 3a) using minimumcompression power as the objective function. Subcooling ofliquid refrigerant is allowed and compression takes place in onestage. External cold utility is available to cool hot streams downto 30 °C. The minimum temperature difference is 5 °C, the

isentropic compression efficiency is 80%, and the physicalproperties calculations are based on the Peng-Robinson equa-tion of state (WORK built-in). Refrigerant expansion occurs inJoule-Thompson valves and the refrigerant pressure drop acrossthe heat exchangers is neglected. The temperature-enthalpy dataof the natural gas are given in Table 1.

6.1. Base Case Solution. The application of the newmethodology results in an optimal solution with a compressionpower of 33.39 MW. The composite curves for this solutionare 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 newmethodology. However, it must be considered that because theminimum temperature difference was not fully enforced, Lee’ssolution ended up with an effective ∆Tmin of approximately 1.2°C. Hence, for the comparison to be really meaningful, the basecase was reoptimized with the new methodology and ∆Tmin of1.2 °C, finding an optimal solution that features 24.53 MW ofpower, that is, 7.8% lower than in Lee’s solution. The threesolutions discussed are reported in Table 2. Note that atemperature approach as tight as 1.2 °C may result as impracticalas it is unlikely to handle the normal operational variations thatmay occur in the plant once constructed. In practice it is unlikelythat temperature approaches of less than 3 °C are consideredduring design. Solving the base case involved the optimizationof 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 notingthat all three solutions presented in the previous section have acompressor outlet temperature of no less than 140 °C. A gasundergoing compression at such temperatures certainly occupiesa relatively high specific volume and hence demands a relativelyhigh specific compression power. To illustrate the effect ofmultistage compression, another design task was performed. Thistime the maximum stage pressure ratio was set to 5 and theminimum temperature difference back to 5 °C. The resulting

Figure 8. Optimization strategy.

Table 1. Temperature-Enthalpy Data of the Natural Gas Stream toBe Liquefied

temperature (°C) enthalpy (MW)

25.0 20179-6.0 18317-34.1 16353-57.7 14468-70.1 11978-74.6 10198-82.3 7114-96.5 5690-115.0 3840-163.0 0

Figure 9. Composite curves for the base case optimal solution.

Table 2. The Three Solutions for the Base Case

new method Lee’s method new method

∆Tmin (°C) 5.0 1.2 1.2power (MW) 33.49 26.60 24.53F1 (kmol/s) 3.47 3.2 3.53PL0

IN (bar) 2.40 3.7 4.84PH0

OUT (bar) 36.95 40.0 43.87Z1 (mol %)N2 15.32 11.0 10.08CH4 17.79 27.3 27.12C2H6 40.85 35.6 37.21C3H8 0.41 5.20 0.27n-C4H10 25.62 20.9 25.31


minimum power solution has total compression power of 27.87MW, which is 16.8% lower than the minimum power foundwhen optimizing for a single compression stage (33.49 MW).Details are shown in Figure 10. The compression train featuresthree stages with a pressure ratio of 2.95 each. The totalintercooler duty was also reduced from 53.67 to 48.15 MW.Although the refrigerant composition is similar to that of thebase case optimal solution, meaningful differences are observedin the refrigerant flowrates (2.81 against 3.47 kmol/s before)and in the compressor inlet and outlet pressures (1.76 against2.40 bar and 45.06 against 36.95 bar). It seems that now thatthe compression process is more efficient, the system can afforda larger total pressure ratio (25.6 against 15.4) in exchange fora reduction in the refrigerant flowrate, the net effect being theobserved reduction in total power while still keeping good careof temperature feasibility in the cryogenic exchanger.

If the compression ratios are not assumed identical butallowed to vary as additional degrees of freedom duringoptimization, the optimal power exhibits a slight furtherreduction to 27.57 MW. In that case the compression ratios forthe three compression stages are 3.57, 2.83, and 2.30, respec-tively, which are approximately within (20% of the 2.95 thatresulted when assumed identical.

One could be tempted to say that a multistage compressionmodel is not needed at the synthesis stage because the powerconsumption can be reduced after optimizing the cycle for asingle compression stage by just increasing the number ofcompression stages afterward. However, doing so in the casestudy reduces the total power only to 30.18 MW; still 2.31 MWabove the 27.87 MW found when considering the effect ofmultiple stages during optimization. One possible reason forthis is the unattractiveness of exploring high total pressure ratioswhen minimizing power in a cycle with a single compressionstage, as discussed in the previous paragraph.

6.3. Two-Stage Refrigeration and Capital Cost Effect. Atwo-stage MR cycle was also optimized in a sensitivity analysiswith a maximum stage pressure ratio of 5. The goal was to

picture the effect that the predefined minimum temperaturedifference has on the minimum power solutions. The effect onthe total compression power can be seen in Figure 11a, wherethe resulting power of each solution is plotted against therespective value of ∆Tmin. The monotonic trend was to beexpected. The closer the hot and cold composite curves areallowed, the lesser the irreversibilities in the process and hencethe lesser the total compression power. Note that the optimiza-tion for a ∆Tmin of 5 °C gave a solution with a powerconsumption of 26.58 MW. This is 1.29 MW lower than thesolution reported in the previous section and due to the factthat the refrigeration stages have been increased to two, whichin this case is helping to achieve a better match between thecomposite curves in each stage.

The capital cost of the solutions appearing in Figure 11a areplotted in Figure 11b also against the minimum temperaturedifference. The capital cost considered for any given solutionincludes the cost of the compressor and the cost of the cryogenicheat exchangers. The latter are estimated using the method inESDU11 for aluminum-brazed plate-fin heat exchangers (PFHE),which estimates the volume and cost of these on the basis ofthe composite curves and typical heat transfer coefficients giventhe type of application. This time the trend is not monotonic. A

Figure 10. Compression train for optimal solution with maximum stage pressure ratio of 5.

Figure 11. ∆Tmin sensitivity for two-staged refrigeration cycles.

Figure 12. Composite curves for the optimal MR system with fourrefrigeration stages.


minimum capital cost seems to exist somewhere between 1 and5 °C. This behavior is also not surprising. As the total powerdecreases 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 largerthe heat transfer area, volume, and cost.

With the purpose of further investigating this compressor/PFHE capital tradeoff, the minimum temperature differencepermitted during optimization was decreased to a negligibleamount and the objective of the optimization was changed tominimum capital cost. In this way there would be no restrictionson the temperature differences, apart from avoiding crossovers,as long as the capital is minimized. The resulting solution isalso shown in Figure 11. In the point of closest approachbetween hot and cold composite curves the temperature differ-ence was 1.3 °C. The total compression power is located slightlyabove the power curve of minimum power solutions. However,the capital cost is considerably below the cost curve of minimumpower solutions. The reason for this is that, apart from exploitingthe compressor/PFHE capital tradeoff, the resulting pressurelevels of the minimum capital solution are such that thecompression task can be accommodated in just two stages,against the three needed in all of the minimum power solutions.

This reduction in the number of compression stages, albeit therebeing the slightly increased total power, provided an additionalcontribution toward the capital minimization due to the economyof scale.

7. Refrigeration Systems in Cascade

In the previous case study, using two stages of refrigerationallowed a 4.6% of reduction in the total compression powerwhen compared to a single refrigeration stage system withmultistage compression. In that case, a better match betweenthe composite curves was achieved because a favorable changeof the refrigerant flowrate and average composition occurred atan intermediate temperature. However, this is not to begeneralized for systems with more refrigeration stages. It is notalways true that the more refrigeration stages there are in anMR system of the type in Figure 5a, the lower is the totalcompression power. In fact, when three refrigeration stages areused in the given example, the minimum power solution requires26.95 MW of compression power, a slight increase over the26.58 MW of the optimal two-stage system, although still lessthan the 27.87 MW of the optimal single-stage system.Furthermore, the lowest possible compression power resultedin 32.94 MW for an MR system with four refrigeration stages,which is considerably more than in the case of one-, two-, andthree-stage systems.

The above behavior is because, with each additional refrig-eration stage, an intermediate separation of the vapor and liquidrefrigerant streams is carried out, and this may introduce anunfavorable change in the refrigerant flowrates and compositionsin the downstream refrigeration stage(s). These changes in turnmay make it difficult to achieve a good match between thecomposite curves because the designer does not have a completecontrol on the former. An illustration of this is shown in Figure12, which corresponds to the optimal four-stage system. Therefrigerant flowrates and compositions in the different stagesare not completely independent from each other. They are verymuch a consequence of the chosen overall refrigerant composi-tion (the one through compression) and the intermediatetemperature between stages.

An alternative to increase the complexity within the samerefrigeration cycle to achieve a more efficient overall system isto employ refrigeration systems in cascade. This is particularly

Figure 13. General cascade of two MR cycles.

Figure 14. Optimization strategy for MR cycles in cascade.


useful for refrigeration over wide temperature ranges and meanssharing the refrigeration duty between two or more cycles, wherethe colder cycle(s) reject heat to the warmer one(s) andeventually the warmest cycle rejects heat to an ambient utility.The process stream(s) release heat to these cycles in decreasingorder of temperature for more efficiency. This is releasing asmuch heat as possible to the warmest cycle, down to the firstso-called partition temperature, then releasing as much heat aspossible to the second warmest cycle, down to the secondpartition temperature, and so on. The cycles within the cascadedo not have to be of the same nature (e.g., pure refrigerant,MR, expander cycles). Cascaded cycles are commonly appliedin the LNG industry.

7.1. Optimal Design of MR Cycles in Cascade. A generalcascade of two MR cycles is shown in Figure 13. The processstream(s), with an inlet temperature TPIN, release heat in theupper (warmer) MR cycle, with NRU refrigeration stages, untilreaching the partition temperature TP, and then undergo(es)further cooling in the lower (colder) MR cycle, with NRL

refrigeration stages, until the required final temperature TPOUT

is reached. After compression, the refrigerant in the lower cyclerejects as much heat as possible to an external cold utility downto 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 first of the NRL stages of the lowercycle, where it undergoes self-cooling, phase splitting, andexpansion as in previously described noncascade MR systems.In the upper cycle, the upper refrigerant rejects heat only to theexternal cold utility down to the temperature TRIN before beingsent to the first of the NRU refrigeration stages. The upperrefrigerant is to provide cooling to the process stream(s) and tothe lower refrigerant.

The approach previously described in detail in the optimaldesign of (noncascade) MR cycles can be adapted to a cascadeMR cycles without much difficulty. The basic idea of theoptimization framework in Figure 8 is still valid in this caseand an adaptation to a cascade cycle is presented in Figure 14.The optimizer proposes a series of candidate solutions to beevaluated by the simulator but this time the degrees of freedomin both the upper and the lower cycles, along with the partitiontemperature, are manipulated by the optimizer. The simulatoris to apply basically the same problem formulation describedpreviously to each cycle in order to evaluate the objectivefunction.

As the stage heat duties in the upper cycle depend on theflowrate and properties of the MR stream received from thelower cycle, the latter is simulated first. After this, the lowerMR cooling curve from TRIN to TP is computed and combinedwith the process composite curve from TPIN to TP to producethe “effective” process composite curve in the upper cycle. Thiseffective curve then defines the actual external refrigerationduties in the upper cycle. The calculation of this lower cyclecooling profile and of the effective process composite curvefor the upper cycle is represented by eq 69-72, which must beadded to the problem formulation in order to solve the uppercycle. The effective composite curve data TPREF and HPREFwould replace TPR and HPR in the rest of the formulation whenapplied to the upper cycle.

TLR)TPROF(TIO, TP, PHLOUT

0, (PHLOUT

0 -∆PLR), ZL1)

(69)

HLR)HPROF(TI0, Tp, PHLOUT

0,(PHLOUT

0 -∆PLR), ZL1, FL1

(70)

TPREF )TCOMP(TPRU, TLR, HPRU, HLR) (71)

HPREF )HCOMP(TPRU, TLR, HPRU, HLR) (72)

Other minor adjustments in the formulation include replacingTPIN and TRIN with TP and PH0

OUT with (PHLOUT

0 - ∆PLR) inthe lower cycle model, except for the equations related tocompression, and TPOUT with TP in the upper cycle model. Theobjective function remains as a unique equation and has to beredefined as a function of the features in both the lower and theupper cycles. The number of stages for both cycles can be treatedas integer variables, and consequently overall optimizationframework can be formulated with mixed-integer nonlinearprogramming (MINLP). However, the optimization has beencarried out with nonlinear programming (NLP) with prespeci-fying the number of stages employed in each cycle, in order toavoid computational difficulties associated with the introductionof integer variables.

8. Case Study 2 - MR Cycles in Cascade

The approach for the optimal design of MR cycles in cascadewill be illustrated by tackling a problem previously publishedas a case study by Vaidyaraman and Maranas.6 A natural gasstream contains 93 mol % of methane, 5 mol % ethane, 1.5mol % propane, and 0.5 mol % n-butane and is to be chilledfrom an inlet temperature of 19.85 to -58.15 °C at a constantpressure of 42 bar. A cascade of two MR cycles, with a variednumber of refrigeration stages each, provides the requiredcooling. The hot liquid refrigerant streams are not subcooledin 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 44arenot required.). Contrary to the principles of an efficient system,the process stream is not precooled in the upper MR cycle butfed directly to the lower MR cycle. The former is only used forheat rejection from the latter. As the intention of this case studyis to compare the results of the new design approach to thosefrom Vaidyaraman and Maranas,6 this flowsheet feature wasleft untouched in order to make the comparison on the samebasis. Considering the above features, the generic flowsheet usedin this case study is shown in Figure 15. A further slightadjustment of the problem formulation is required to model sucha process, the process stream(s) will not contribute to theeffective process composite curve in the simulation of the uppercycle (i.e., TPRU and HPRU do not participate in eqs 71 and72) and second, the process stream(s) enter the lower cycle atthe temperature TPIN instead of the partition temperature TP.

The refrigerants in the lower and the upper cycles containethane, propane and n-butane. After compression in a singlestage, these reject heat to an external cold utility until reachinga temperature of 36.85 °C. The pressure drop of the refrigerantsthrough the system is neglected and the minimum temperaturedifference allowed in the main heat exchangers is 2.5 °C. TheSoave-Redlich-Kwong equation of state provides the basisfor physical property calculations, which for this case study areperformed by interfacing with Aspen Properties.

The objective of the optimization is to minimize the coef-ficient of performance of the system for different combinationsin the number of lower and upper refrigeration stages. Thecoefficient of performance (COP) in this case is defined as theratio between the total compression work and the amount ofheat released by the process stream. According to this definition,the lower the COP the more efficient the system. If an arbitraryflowrate for the process stream is chosen, then minimizing COPwould in practice be the same as minimizing total compressionpower as the amount of heat released by the process is fixed.


The pressures of the refrigerants at the compressor inlet areallowed to vary between 1.0 and 2.5 bar, the discharge pressuresbetween 4 and 10 bar, the compression ratios between 1.5 and12 (but only a single compression stage) and the refrigerantflowrates between 0.1 and 1.0 kmol/s, which is adequate for anassumed 1 kmol/s of natural gas. The partition temperature isallowed to vary between -40 and 10 °C when a cascade ispresent.

The optimal COPs encountered when applying the proposedapproach are presented in Table 3 for different combinationsof the number of stages in the lower cycle (NRL) and those inthe upper cycle (NRU). Details on each of these solutions canbe found in the Appendix. The COPs obtained with the previousapproach6 are shown in Table 4 for comparison purposes.

For more simplicity, Table 5 shows the relative improvementsin the system performance (COP) achieved with the newapproach. In only three cases it was possible to obtain lowerCOPs, with the best saving over the previous method being8.09%, in the case of one stage in the lower cycle and one inthe upper cycle. In most cases applying the new method resultedin higher COPs, by up to 16.48%. However, contrary to whatone could initially think, this is evidence of the strength of thenew method rather than a weakness, as all the new solutions

are feasible ones. As discussed previously in the literaturereview, the method by Vaidyaraman and Maranas6 enforcedtemperature feasibility only at the ends of each refrigerationstage. As a result, temperature violations are plentiful in theprevious results and an increase in the total power seems to bethe only way to regain feasibility.

For instance, assuming 1 kmol/s of natural gas, Figure 16shows the composite curves of the lower and upper cycles thatcorrespond to the previous optimal solution for a system withthree stages in the lower cycle and three in the upper cycle.This is the same case in which the new method yields the highestrelative COP increase as in Table 5. While the upper cycleexhibits only a slight ∆Tmin violation at the cold end of thesecond stage, this violation is quite significant in the second

Figure 15. Generic flowsheet for case study 2.

Table 3. Optimal COPs Obtained with the New Approach

NRU NRL ) 0 (no cascade) NRL ) 1 NRL ) 2 NRL ) 3

1 - 0.5415 0.5024 0.45662 0.5291 0.4485 0.4755 0.49513 0.5238 0.4266 0.4263 0.48054 0.5488 0.4790 0.4581 -

Table 4. Optimal COPs obtained by Vaidyaraman and Maranas6


1 - 0.5890 0.4608 0.46232 0.5022 0.4547 0.4095 0.42983 0.4562 0.4144 0.3957 0.41244 0.4948 0.4226 0.4298 -

Table 5. Relative COP with the New Approach (100 ×COPNEW/COPPREV)


1 - 91.91 109.00 98.732 105.33 98.61 116.08 115.173 114.80 102.85 107.77 116.484 110.88 113.31 106.55 -

Figure 16. Composite curves of the previous solution for three stages inthe lower cycle and three in the upper cycle.


stage of the upper cycle. Furthermore, a slight temperaturecrossover occurs.

On the other hand, the composite curves of the optimalsolution found with the new approach are those in Figure 17,showing no crossovers or minimum temperature approachviolations. A more detailed comparison of both solutions ispresented in Table 6, with meaningful differences found in mostvariables, contrary to the suggestion by Vaidyaraman andMaranas6 that any resulting infeasibility could be rectified withslight changes in pressure and still keep optimality. In generalthe new solution (1) allocates a larger share of refrigeration tothe lower cycle than the previous solution (the partitiontemperature is now more than 14 °C lower), (2) features moremethane and less propane and n-butane in the refrigerantmixtures, and (3) is able to afford a reduction in the refrigerant

flowrates in exchange for increased compression ratios. The neteffect is, as previously mentioned, a 16.48% of increase in theCOP (and total power) of the system but, nevertheless, a fullcompliance with temperature feasibility.

The size of the problems solved in this case study rangedbetween 5 and 16 variables subject to optimization, each casetaking between 77 and 295 min to solve in a Pentium IVprocessor (3.0 GHz) with 512 Mb of RAM. Typical GAparameters used in the optimization are population size )1000, number of generations ) 600, crossover probability) 0.85, mutation method ) one-point mutation with adjust-able rate based on fitness, relative fitness differential ) 0.5,steady-state-delete-worst reproduction, elitism technique isapplied.

9. Conclusions

The design of MR cycles is a challenging task. Also, localoptima are abundant. This paper has described and illustratedthe application of a new formulation and optimization strategyfor the synthesis of MR cycles. It considers multistagerefrigerant compression, full enforcement of the minimumtemperature difference along the temperature profiles, simul-taneous optimization of variables, incorporation of capitalcosts in the objective function and the use of stochasticoptimization (genetic algorithm) to overcome local optima.The effectiveness of the method was illustrated by findingimproved and feasible solutions for two previously publishedcase studies. The incorporation of multistage compressionproved a key factor in the design of more efficient cyclessince it not only reduces the power consumption per se, butalso encourages the full exploration of total pressure ratios,unlike the single stage case. Although only compressor andPFHE capital costs were considered, the tradeoff betweenthese was illustrated, and how in practice this means thatthe cost of power-related items is dominant in this type ofsystems. When applied to MR cycles in cascade, in somecases it was possible to obtain new designs with improvedthermodynamic performance against the results published inprevious work. In most cases, however, the new solutionsdid not apparently exhibit a performance improvement butnevertheless they were fully compliant with temperaturefeasibility checks, unlike the previous work that enforcedthis only at the ends of the heat exchangers and, as aconsequence, intermediate ∆Tmin violations and sometimestemperature crossovers occurred. Although it makes thesolving process time-consuming and it still does not guaranteeglobal optima, the application of a genetic algorithm opti-mizer to this design problem permits accepting the optimalityof the results with greater confidence than deterministicmethods.

Acknowledgment

First author of this paper would like to thank the Centre forProcess Integration at the University of Manchester and theOverseas Research Student Award Scheme for the financialsponsorship of this project.

Appendix

Details on each of solutions found in case study 2 are listed inTables A1-A4. Refrigerant flowrates are per 1 kmol/s of naturalgas.

Figure 17. Composite curves of the new solution for three stages in thelower cycle (a) and three in the upper cycle (b).

Table 6. Comparison of Solutions for Three Stages in the LowerCycle and Three in the Upper Cycle

previous approach new approach

TP (°C) 14.59 0.32FL1 (kmol/s) 0.499 0.396PLL

IN0 (bar) 2.50 2.36

PHLOUT

0 (bar) 7.00 7.02ZL 1 mol %C2H6 32.80 46.62C3H8 33.34 29.58n-C4H10 33.86 23.80TI L (°C) -11.14, -35.01 -29.09, -51.17

FU 1 (kmol/s) 0.552 0.4747PLU

IN0 (bar) 2.50 2.50

PHUOUT

0 (bar) 4.20 5.22ZU 1 mol %C2H6 0.12 15.98C3H8 12.76 6.81n-C4H10 87.12 77.21TI U (°C) 30.85, 16.53 18.07, 3.44


Nomenclature

Subscriptsi ) compression stage indexL ) lower refrigeration cyclen ) refrigeration stage indexU ) 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 approachηc ) isentropic efficiency of compressorηLEX ) isentropic efficiency of liquid expanderηp ) isentropic efficiency of pumpHPR ) set of enthalpy flows defining the overall process composite

curveNR ) number of refrigeration stagesPRMAX ) maximum pressure or compression ratioPRMIN ) minimum pressure or compression ratioTPIN ) process inlet temperature to the systemTPOUT ) process outlet temperature from systemTPR ) temperatures in the overall process composite curveTRIN ) temperature of hot streams after heat rejection to air or

cooling waterYLEX ) binary parameter indicating the use of liquid expanders

(when set to 1)FunctionsCROP ) extracts a set of enthalpy flows or temperatures contained

within a given temperature range from a larger arrayHCOMP ) combines stream data and returns the set of enthalpy

flows defining a hot composite curveHEV ) returns the enthalpy flow that corresponds to a sample

temperature in a composite curveHPROF ) returns a set of enthalpy flows defining the composite

curve of a stream with the given flowrate and compositions andwithin 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-defined objective function. may change from case tocase, including economic and/or performance criteria

Table A1. Solutions with One Refrigeration Stage in the LowerCycle

refrigeration stages in upper cycle

variable 0 (no cascade) 1 2 3

TP (°C) - -21.78 -16.39 -44.69FL1 (kmol/s) - 0.2019 0.2187 0.1834PLL

IN0 (bar) - 1.398 1.533 1.911

PHLOUT

0 (bar) - 7.064 9.588 9.069ZL 1 mol %C2H6 - 0.5069 0.5297 0.5649C3H8 - 0.1878 0.2843 0.0474n-C4H10 - 0.3053 0.1860 0.3877TI L (°C) - - -

FU 1 (kmol/s) - 0.2381 0.3414 0.3703PLU

IN0 (bar) - 1.100 2.109 2.500

PHUOUT

0 (bar) - 8.849 6.421 8.773ZU 1 mol %C2H6 - 0.0085 0.0546 0.1913C3H8 - 0.5701 0.3353 0.2599n-C4H10 - 0.4214 0.6102 0.5488TI U (°C) - - 4.71 3.03, -26.50

Table A2. Solutions with Two Refrigeration Stages in the LowerCycle



TP (°C) - 3.29 -6.82 -5.92FL1 (kmol/s) 0.4138 0.3445 0.3529 0.3209PLL

IN0 (bar) 1.421 2.261 1.675 2.494

PHLOUT

0 (bar) 9.618 7.980 4.623 8.310ZL 1 mol %C2H6 0.3636 0.4244 0.3241 0.5263C3H8 0.0870 0.3255 0.4566 0.2639n-C4H10 0.5494 0.2501 0.2193 0.2098TI L (°C) -15.45 -29.52 -37.63 -36.24

FU 1 (kmol/s) - 0.2106 0.3804 0.5236PLU

IN0 (bar) - 1.243 2.499 2.477

PHUOUT

0 (bar) - 4.610 7.613 5.674ZU 1 mol %C2H6 - 0.0216 0.0067 0.1759C3H8 - 0.0468 0.5384 0.0973n-C4H10 - 0.9315 0.4549 0.7268TI U (°C) - - 3.60 18.73, 0.06

Table A3. Solutions with Three Refrigeration Stages in the LowerCycle



TP (°C) - 0.18 5.04 0.32FL1 (kmol/s) 0.5821 0.4430 0.4504 0.3959PLL

IN0 (bar) 2.490 2.479 2.428 2.363

PHLOUT

0 (bar) 9.935 5.800 6.140 7.024ZL 1 mol %C2H6 0.3797 0.3638 0.3617 0.4662C3H8 0.1833 0.4399 0.3942 0.2958n-C4H10 0.4369 0.1963 0.2441 0.2380TI L (°C) 3.26,

-30.04-20.87,-40.75

-19.40,-40.91

-29.09,-51.17

FU 1 (kmol/s) - 0.2548 0.3493 0.4747PLU

IN0 (bar) - 2.500 2.469 2.500

PHUOUT

0 (bar) - 8.601 5.519 5.224ZU 1 mol %C2H6 - 0.0076 0.0259 0.1598C3H8 - 0.5412 0.2717 0.0681n-C4H10 - 0.4512 0.7024 0.7721TI U (°C) - - 14.18 18.03, 3.44

Table A4. Solutions with four refrigeration stages in the lower cycle



TP (°C) - -11.67 11.55 -FL1 (kmol/s) 0.6337 0.5926 0.5060 -PLL

IN0 (bar) 2.461 2.499 2.499 -

PHLOUT

0 (bar) 9.690 4.397 7.193 -ZL 1 mol %C2H6 0.3820 0.3902 0.4154 -C3H8 0.1626 0.4723 0.2893 -n-C4H10 0.4554 0.1375 0.2953 -TI L (°C) 10.60, -21.08,

-50.51-24.20, -39.60,-50.86

-12.61, -33.05,-50.58

-

FU 1 (kmol/s) - 0.3140 0.3239 -PLU

IN0 (bar) - 1.743 2.451 -

PHUOUT

0 (bar) - 9.042 4.845 -ZU 1 mol %C2H6 - 0.0000 0.0322 -C3H8 - 0.6131 0.1570 -n-C4H10 - 0.3869 0.8108 -TI U (°C) - - 17.58 -


sTP ) returns the entropy of a stream given its composition,temperature and pressure

TCOMP ) combines stream data and returns the temperatures ina hot composite curve

TDEW ) returns the dew temperature of a stream given itscomposition and pressure

TEV ) returns the temperature that corresponds to a sampleenthalpy flow in a composite curve

ThP ) returns the temperature of a stream given its composition,enthalpy and pressure

TPROF ) returns a set of temperatures defining the composite curveof a stream with the given flowrate and compositions and withinthe 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 vaporisedstream given its total composition, temperature and pressure

ZLIQ ) returns the liquid composition of a partially vaporisedstream given its total composition, temperature and pressure

VariablesF ) total molar flowrate of hot refrigerant entering a given

refrigeration stageFC ) total molar flowrate of cold refrigerant entering a given

refrigeration stageFCMP ) molar flowrate of refrigerant entering a given compression

stageFL ) liquid molar flowrate of hot refrigerant entering a given

refrigeration stageFP ) liquid molar flowrate of refrigerant after intercooler in a given

compression stageFV ) vapor molar flowrate of hot refrigerant entering a given

refrigeration stageHC ) set of enthalpy flows defining the composite curve of the

cold refrigerant in a given refrigeration stagehCIN ) cold refrigerant enthalpy at the inlet of a given refrigeration

stageHCCOMP ) set of enthalpy flows defining the composite curve of

the cold streams in a given refrigeration stagehCMP ) enthalpy of refrigerant after a given compression stagehCOUT ) cold refrigerant enthalpy at the outlet of a given

refrigeration stagehEXPN ) enthalpy of refrigerant after expansion in valve or liquid

turbine in a given refrigeration stageHHCOMP ) set of enthalpy flows defining the composite curve of

the hot streams in a given refrigeration stageHHL ) set of enthalpy flows defining the composite curve of the

liquid hot refrigerant stream in a given refrigeration stageHHV ) set of enthalpy flows defining the composite curve of the

vapor hot refrigerant stream in a given refrigeration stageHLR ) set of enthalpies defining the composite curve of the lower

refrigerant through the upper cyclehP ) enthalpy of refrigerant after pumping in a given compression

stageHP ) set of enthalpy flows defining the composite curve of the

process stream(s) in a given refrigeration stageHPREF ) set of enthalpy flows defining the effective process

composite curve in the upper cyclehISCMP ) isentropic enthalpy of compression in a given compres-

sion stagehISLEX ) isentropic enthalpy of expansion of cold liquid refrigerant

in a given refrigeration stagehISP ) isentropic enthalpy of pumping in a given compression stageNC ) number of compression stages

ObjectiVe) objective function valuePLIN ) pressure of cold refrigerant at the inlet of a given

refrigeration stagePHOUT ) pressure of hot refrigerant at the outlet of a given

refrigeration stagePR ) pressure ratio of a given compression stage when dissimilar

values are allowed. unique stage pressure ratio otherwisePCMP ) discharge pressure in a given compression stageQC ) heat received by the cold refrigerant in a given refrigeration

stageQCMP ) heat removed in intercooler after a given compression

stageQL ) heat removed from the hot liquid refrigerant in a given

refrigeration stageQP ) heat removed from the process stream(s) in a given

refrigeration stageQV ) heat removed from the hot vapor refrigerant in a given

refrigeration stageTC ) set of temperatures defining the composite curve of the cold

refrigerant in a given refrigeration stageTCCOMP ) set of temperatures defining the composite curve of

the cold streams in a given refrigeration stageTCIN ) cold refrigerant temperature at the inlet of a given

refrigeration stageTCOUT ) cold refrigerant temperature at the outlet of a given

refrigeration stageTEXPN ) temperature of refrigerant after expansion in valve or

liquid turbine in a given refrigeration stageth ) sample temperature for feasibility evaluationTHCOMP ) set of temperatures defining the composite curve of

the hot streams in a given refrigeration stageTHL ) set of temperatures defining the composite curve of the

liquid hot refrigerant stream in a given refrigeration stageTHV ) set of temperatures defining the composite curve of the

vapor hot refrigerant stream in a given refrigeration stageTI ) temperature of the hot streams after a given refrigeration stage

(intermediate temperature)TLR ) set of temperatures defining the composite curve of the

lower refrigerant through the upper cycleTP ) set of temperatures defining the composite curve of the

process stream(s) in a given refrigeration stageTPREF ) set of temperatures defining the effective process

composite curve in the upper cycleWLEX ) work produced in liquid turbine at a given refrigeration

stageWC ) work required in a given compression stageWP ) pumping work required after a given compression stageX ) molar composition of hot liquid refrigerant in a given

refrigeration stageXCMP ) molar composition of liquid after intercooler in a given

compression stageY ) molar composition of hot vapor refrigerant in a given

refrigeration stageYCMP ) molar composition of vapor after intercooler in a given

compression stageZ ) molar composition of cold refrigerant in a given refrigeration

stage

Literature Cited

(1) Walsh, B. W. Mixed refrigerant process design. 21st AustralasianChemical Engineering Conference, 1993, 1, 59/1-64/1.

(2) Finn, A. J.; Johnson, G. L.; Tomlinson, T. R. Developments inNatural Gas Liquefaction. Hydrocarbon Processing 1999, 78 (4), 47.


(3) Ait-Ali, M. A. Optimal mixed refrigerant liquefaction of natural gas.Ph.D. Thesis. Stanford University, CA, 1979.

(4) Lee, G. C. Optimal design and analysis of refrigeration systems forlow temperature processes. Ph.D. Thesis. UMIST. U.K., 2001.

(5) Lee, G. C.; Smith, R.; Zhu, X. X. Optimal Synthesis of Mixed-refrigerant Systems for Low Temperature Processes. Ind. Eng. Chem. Res.2002, 41, 5016.

(6) Vaidyaraman, S.; Maranas, C. D. Synthesis of Mixed RefrigerantCascade Cycles. Chem. Eng. Commun. 2002, 189 (8), 1057.

(7) Pua, L. M.; Zhu, X. X. Integrated Heat Exchanger Network andEquipment Design Using Compact Heat Exchangers. Heat Transfer Eng.2002, 23 (6), 18.

(8) Goldberg, D. E. Genetic Algorithms in Search, Optimization andMachine Learning; Addison-Wesley: Reading, MA, 1989.

(9) Stender, J.; Hillebrand, E.; Kingdon, J. Genetic Algorithms inOptimisation, Simulation and Modeling; IOS Press: Amsterdam, TheNetherlands, 1994.

(10) Charbonneau, P. Release Notes for PIKAIA 1.2, National Centerfor Atmospheric Research, Technical Note 451+STR, 2002.

(11) Engineering Science Data Unit International PLC. Selection andCosting of Heat Exchangers-Plate-Fin Type; ESDU: London, 2003.

ReceiVed for reView April 2, 2008ReVised manuscript receiVed August 4, 2008

Accepted August 12, 2008

IE800515U


Documents

Optimal Design of Mixed Refrigerant Cycles