optimal synthesis of mixed-refrigerant systems for low-temperature processes

Optimal Synthesis of Mixed-Refrigerant Systems forLow-Temperature Processes

G. C. Lee,† R. Smith, and X. X. Zhu*

Department of Process Integration, The University of Manchester Institute of Science and Technology,P.O. Box 88, Manchester M60 1QD, U.K.

Using mixtures as refrigerants in the design of refrigeration systems offers significantopportunities in the search for more energy-efficient and compact designs. However, the designof mixed-refrigerant systems is extremely difficult, and few successful design methods areavailable. As a result, many existing operations can be far from the optimal conditions. In thispaper, a novel method for the selection of refrigerant compositions has been proposed. Usingthis approach, a systematic synthesis method for the complete design of mixed-refrigerantsystems has been developed. This approach combines the power of thermodynamics and nonlinearprogramming (NLP) techniques. Whereas NLP can determine the optimal choice of processoperating conditions, thermodynamics simultaneously provides insights into and confidence inthe solution. More importantly, the use of thermodynamic analysis in this approach helpsovercome the complexity in NLP optimization for determining key design variables, includingrefrigerant composition, operating pressures, and flow rates. Case studies demonstrate that themethod is effective and that mixed-refrigerant systems can significantly save energy and improveprocess efficiency.

1. Introduction

Refrigeration systems are used extensively in manyindustrial processes. Especially in low-temperatureprocesses, such as ethylene recovery or gas liquefaction,refrigeration systems play a critical role in the overallenergy consumption and capital costs. Therefore, thedesign of more energy-efficient and low-cost refrigera-tion systems provides a great impact on plant operationand economics.

A mixed-refrigerant (MR) system uses a mixture asthe refrigerant rather than several pure refrigerants asin conventional multistage or cascading refrigerationsystems. Unlike pure refrigerants and azeotropic mix-tures, nonazeotropic mixtures do not maintain a con-stant temperature and vapor and liquid composition atconstant pressure as the refrigerants evaporate orcondense. The composition of the mixture is selectedsuch that the liquid refrigerant evaporates over atemperature range similar to that of the process coolingdemand. A mixture of hydrocarbons (usually in the C1-C5 range) and nitrogen is normally used to provide thedesired refrigerant characteristics (e.g., close matchingof the hot and cold composite curves, with small tem-perature driving forces over the whole temperaturerange) for the specific refrigeration demand. The smalltemperature driving force leads to near-reversible op-eration and, thus, better thermodynamic efficiency andalower power requirement. Also, an MR system featuresa simpler machinery configuration and fewer mainte-nance problems.

The concept of using a mixture as a refrigerant hasbeen around for a long time. The earliest applicationcan be traced to 1936, when W. J. Podbielniak1 devised

a mixed-component refrigerant system having threestages of throttling and a single compressor in a closedcycle. Since then, several variations and applicationshave been introduced.2-4 MacKenzie and Donnelly5

demonstrated that MR systems are more efficient thanturbo-expander systems in natural gas liquid (NGL)recovery processes. More recently, because of the grow-ing concern of the impact of ozone depletion by certainchlorofluorocarbon (CFC)-based refrigerants, there is anurgent need to replace refrigerants with high ozonedepletion potentials (ODPs) by environmentally benignones.6 Using mixed refrigerants is one of the mostpromising solutions.7 Lamb et al.8 and Bensafi andHaselden9 demonstrated that MR systems can achievehigh energy efficiency and thus power savings. Theyreported up to 30% energy savings compared withsystems using pure R22. However, they determined thebest composition of the refrigerant mixture experimen-tally.

Duvedi and Achenie10 used an MINLP approach forthe design of refrigerant mixtures that have the desiredattributes, such as low ODPs. However, the approachwas limited to a small number of refrigerant compo-nents, and the assumptions made in the MINLP modelwere far from realistic. First, they assumed that therefrigerant mixtures leaving the condenser or theevaporator were at their saturated states. However, thisassumption is untrue, because, in reality, the refrigerantmixtures leaving the condenser should be subcooled andthe mixtures leaving the evaporator should be super-heated. Second, they assumed that the evaporating andcondensing temperatures were the average of the inletand outlet saturated temperatures. In practical opera-tions, the evaporating and condensing temperatures ofrefrigerant mixtures usually cover a wide range fromambient to -160 °C. For such a wide temperature range,the second assumption is poor. In the design of MRsystems, we need to be concerned not only with theminimization of energy and capital costs, but also with

* Corresponding author. E-mail: [email protected]. Currentaddress: UOP LLC, 25 East Algonquin Rd., Des Plaines, IL60017-5017.

† Current address: AspenTech Inc., 1293 Eldridge Parkway,Houston, TX 77077. E-mail: [email protected].

5016 Ind. Eng. Chem. Res. 2002, 41, 5016-5028

10.1021/ie020057p CCC: $22.00 © 2002 American Chemical SocietyPublished on Web 08/30/2002

the temperature profiles of the evaporation and con-densation processes. Usually, the temperature approachwithin the heat exchangers of MR systems is as smallas ∼1-3 °C.

Patel and Teja11 proposed a better equation of state(EOS) for predicting the thermodynamic properties ofrefrigerant mixtures. Lee et al.12 improved the Patel-Teja EOS to be a purely predictive model for mixtureproperty calculations. In this work, the Peng-RobinsonEOS will be used, mainly because it is well-developedand widely recognized.

One major application of the MR systems is in theliquid natural gas (LNG) industry.13 Gas liquefactionprocesses are always capital-intensive, because of theextensive use of equipment and the large energy re-quirement in refrigeration. A large part of the invest-ment is in the liquefier, which usually makes up around25-50% of the total cost. Thus, the optimal design andoperation of the liquefaction process offers huge poten-tial energy and cost benefits. There are two main typesof refrigeration systems: cascade refrigeration systemsand MR systems. In fact, the concept of using a mixtureas a refrigerant arose from the awareness of thedifficulty in maintaining classical cascade refrigerationmachines. Because of the wide temperature range in theLNG process, usually from ambient temperature toaround -160 °C, cascade refrigeration systems typicallyrequire three different multistage refrigeration cycles.Each cycle typically comprises three stages, as shownin Figure 1. On the other hand, an MR system can haveonly one compression train and a simpler machineryconfiguration, as shown in Figure 2.

An efficient process can be designed by the judiciousmanipulation of the operating pressure and the com-position of the circulating mixed refrigerant and by theproper arrangement of the heat exchangers. Because therefrigerant evaporation temperature profile can bealtered by changing the refrigerant composition, MRsystems have lower heat-transfer irreversibilities withinheat exchangers, thus saving power. Furthermore, bychanging the refrigerant composition, more efficientutilization of the total available heat-transfer surfaceand the actual compressor characteristics can beachieved.

The PRICO process,14 depicted in Figure 3, is thesimplest form of MR system.

Its major function is the conversion of natural gas(NG) to the liquid state for transportation and/or storageat atmospheric pressure using a single-stage MR sys-tem. It is also applicable to natural gas liquid (NGL)extraction (high ethane recovery) from NG. A mixedrefrigerant is compressed and passed through the mainheat exchanger, where it is condensed. It is thenexpanded across a Joule-Thomson valve and returnedcountercurrently through the heat exchanger to thecompressor. Natural gas enters the heat exchanger atambient temperature and exits as liquefied natural gas.The main heat exchanger is a plate-and-fin apparatuswith a brazed aluminum core. Figure 4 shows thetemperature profiles of the three streams in the heatexchanger: the NG-LNG stream, the warm refrigerantstream (before the Joule-Thomson valve), and the coldrefrigerant stream (after the Joule-Thomson valve).

The NG-LNG stream and the warm refrigerantstream are combined as a hot composite curve, while

Figure 1. Cascade refrigeration system for LNG.

Figure 2. Two-stage MR system for LNG.

Figure 3. PRICO process: single-stage MR system.

Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 5017

the cold refrigerant stream alone forms the cold com-posite curve. We denote two ends in the compositecurves as the high-pressure (HP) end and the low-pressure (LP) end. To achieve higher efficiency, thedesign of the MR systems can be evolved to morecomplex schemes, such as multistage MR systems,propane-precooled MR systems, or cascade MR systems.

However, in practice, MR systems are generallyconsidered to have lower efficiencies than conventionalcascade cycles,15 because even though the temperaturedriving force is smaller, the circulation flow of therefrigerant is much higher. The main reason is theusually poor matching between the hot and cold com-posite curves of the existing plants, which means thata higher flow rate of refrigerant is required to avoidtemperature crosses occurring within the heat ex-changer. The means of improving the performance ofMR system therefore lies in better selection of therefrigerant composition. However, because of the highcomplexity of the problem, the selection of refrigerantcompositions has been done by trial-and-error andguided only by heuristics.

In this paper, we propose a novel methodology for thesystematic synthesis of MR systems by a combinedmathematical programming/thermodynamic approach,which can generate optimal design solutions and goodunderstanding of problems. The basic idea is to try toidentify a set of refrigerant compositions that canprovide the best match between the hot and coldcomposite curves at given pressure levels (condensingand evaporating pressures) and refrigerant flow rate.If the search is successful, then the pressure levels and/or the refrigerant flow rate are reduced progressively,and the procedure of finding the best-matching refriger-ant composition is repeated iteratively. The procedureterminates when no set of valid refrigerant compositionscan be found. In other words, there are always temper-ature crosses inside the heat exchanger, so that nofurther improvement is possible. We propose threedifferent forms of objective function: minimization ofthe crossover, minimization of the sum of the crossovers,and minimization of the shaftwork requirement. It hasbeen found that each objective function has its strengthsin different situations, and better optimization resultscan be obtained by using different objective functionsduring one optimization task.

To test the viability of this methodology, three casestudies on a PRICO process were performed. The firstcase demonstrates a 21.3% shaftwork savings by thenew method compared to the commercial PRICO pro-cesses. The second case study, by switching to differentobjective functions during optimization, achieves a 25%savings in shaftwork consumption. The third case studyinvestigates the effect of using different degrees of

temperature shifting on the change of shaftwork re-quirement.

2. Pure Refrigerant vs Mixed Refrigerant

Conventional refrigeration systems, domestic andindustrial, employ single-component refrigerants. Inmost situations, the temperature of the heat source orsink varies during the heat-transfer process, eventhough the evaporating and condensing temperaturesof pure refrigerants are constant. As a result, there areinevitably pinch points in the evaporators or condensers,as seen in Figure 5.

A large temperature difference at one end of the heatexchanger leads to irreversibility, which, in turn, re-duces the efficiency of the refrigeration system. Also, acertain pure refrigerant might be suitable for theexisting operating conditions. However, once the operat-ing conditions change, another pure refrigerant mightbe more desirable. A possible solution would be toreplace the existing refrigerant with a new one, but thisis not practical. Finally, if the difference between thecondensing and evaporating temperatures becomeslarger, the pressure ratio across the compressor in-creases, resulting in an increase in the consumption ofshaftwork.

The problem of using pure refrigerants mainly stemsfrom the fact that the thermodynamic properties of apure refrigerant, the operating conditions imposed bya refrigeration system, and the particular applicationmight not match well with each other. When a mixedrefrigerant is used, this problem can be overcome byselecting the composition of the refrigerant mixture orvariations in the design of the refrigeration system.

According to the Gibbs phase rule, the number ofdegrees of freedom of a system having C components,M independent reactions, and Q phases is calculatedas

Thus, the specification of C - M - Q + 2 intensivevariables of the individual phases completely fixes thethermodynamic state of each phase. For pure-refriger-ant systems, the number of degrees of freedom is

We have only either temperature or pressure to chooseas our degree of freedom. For an MR system of Ccomponents, the number of degrees of freedom becomes

Figure 4. Composite curves for the PRICO process.

Figure 5. Temperature profiles within an evaporator when usinga pure refrigerant.

F ) C - M - Q + 2 (1)

F ) C - M - Q + 2 ) 1 - 0 - 2 + 2 ) 1 (2)

F ) C - M - Q + 2 ) C - 0 - 2 + 2 ) C (3)

5018 Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002

This suggests that, at a given pressure, the boiling (ordew) temperature of the mixed refrigerant will be afunction of the composition and, hence, there are moredegrees of freedom than in pure-refrigerant systems. Ifthe gap between the condensing and evaporating tem-peratures becomes larger, then for pure refrigerantsystems, the difference between the condensing andevaporating pressures must be increased as well, whereasfor MR systems, the refrigerant composition can simplybe altered while the pressure levels are kept the same.This greatly enhances a refrigeration system’s flexibilityand saves a significant amount of shaftwork whenoperating conditions change.

The major difference between pure refrigerants andmixed refrigerants is the shape of the temperatureprofile during the gas-liquid phase transition, as il-lustrated in Figure 6. Therefore, by using a suitablerefrigerant mixture, the average temperature differencecan be made closer to the minimum temperature dif-ference, thus reducing the heat-transfer losses comparedto those obtained with a pure refrigerant. Changing therefrigerant composition, flow rate, evaporating/condens-ing pressures, or configuration of the heat exchangercan alter the shape of the refrigerant evaporating linefor a mixed refrigerant. If the operating conditionschange in the process, it is possible to maintain desir-able performance by adjusting these operating variables.

Because very small temperature differences betweenthe hot and cold composite curves are involved andcomplex phase equilibrium calculations are needed, itis necessary to use rigorous thermodynamic propertycalculations to obtain the accurate information neces-sary. This not only increases the difficulty of modelingfor the problem, but also adds to the nonlinearity in theoptimization. The design of MR systems usually exploitsmultistream heat exchangers, which is also a challeng-ing problem in itself. Figure 7 shows a temperaturecross between hot and cold composite curves.

To guarantee valid heat transfer, we must avoidtemperature crosses in the heat exchanger. The finalconsideration is the water in the inlet stream of thecompressor. The mixed-refrigerant flow, after beingevaporated in the heat exchanger, becomes the inlet

stream to the compressor. It would be damaging tocertain types of compressors if the inlet stream con-tained some amount of water. This situation is il-lustrated in Figure 8.

In conclusion, the desired hot and cold compositecurves, as shown in Figure 9, should have the followingfeatures: (1) The hot and cold composite curves shouldbe close and parallel to each other; thus, heat transfercan be carried out with a near-constant temperaturedriving force distribution. (2) No temperature crossesshould occur. (3) the vapor should be superheated at theend of cold composite curve.

Figure 6. Comparison between using pure refrigerants and mixedrefrigerants.

Figure 7. Temperature cross between hot and cold compositecurves.

Figure 8. Composite curves when compressor inlet streamcontains water.

Figure 9. Composite curves for the ideal conditions.


3. Characteristics of Key Design Variables

In the design of MR systems, three key variables playdominant roles in affecting the overall performance.These are the pressure levels of condensation andevaporation, the refrigerant flow rate, and the refriger-ant composition. Here we discuss the individual char-acteristics of each of those variables.

Pressure Levels. In the MR refrigeration cycle, themixed refrigerant evaporates and condenses at twopressure levels, while both the evaporating and con-densing temperatures vary over a wide range. In thePRICO process as the base case, which is shown inFigure 10a, the condensing and evaporating pressurelevels are at 42 and 3.4 bar, respectively. When the hotand cold composite curves are checked, there is atemperature cross at the HP end of the heat exchanger.To avoid the temperature cross, the difference betweenthe two pressure levels can be increased such that thecondensing and evaporating pressures are 48 and 3.4bar, respectively. As shown in Figure 10b, the temper-ature cross can be avoided by widening the gap for theHP end, but at the same time, the shaftwork require-ment is increased.

Another significant point is that changing the LP endpressure has a large effect on the LP end temperature,while changing the HP end pressure has a large effecton the HP end temperature difference. However, theshaftwork requirement will inevitably increase if thedifference between the two pressure levels is increased.

Refrigerant Flow Rate. Increasing the refrigerantflow rate can widen the gap between the hot and coldcomposite curves. We use the same base case to dem-onstrate the effect of the refrigerant flow rate. As shownin Figure 11a, where the refrigerant flow rate of thebase case is 3.2 kmol/s, the hot and cold compositecurves cross at the HP end. By increasing the refriger-ant flow rate to 3.5 kmol/s, the temperature cross canbe avoided.

However, the shaftwork requirement is also in-creased. It should be noted that, if the refrigerant flowrate is too low, it is difficult, if not impossible, to avoidtemperature crosses in the heat exchanger. Equally, ifthe refrigerant flow rate is too high, there is a greatpotential that a certain amount of water exists in theinlet stream to the compressor. Therefore, the refriger-ant flow rate can be changed only within a range.Increasing the refrigerant flow rate also inevitablyincreases the shaftwork requirement.

Refrigerant Composition. A typical MR systemusually employs more than four different componentsin the refrigerant mixture. According to the Gibbs phaserule, we have C degrees of freedom in MR systems. Byintroducing new components or replacing an existingcomponent by a new one, more degrees of freedom areavailable to adjust to achieve better performance of theMR systems. Consider the same base case to demon-strate the effect of changing the refrigerant composition.In Figure 12a, we face the same temperature crossproblem.

Figure 10. Effect of changing pressure levels.Figure 11. Effect of changing refrigerant flow rate.


By applying the method for selecting the optimalrefrigerant composition that will be introduced later, weobtain another set of refrigerant compositions thatsuccessfully avoid the temperature cross problem anddecrease the shaftwork requirement at the same time.Unlike the previous two characteristics, it is not inevi-table that removing a temperature cross results inhigher shaftwork consumption. Because the either pres-sure levels and refrigerant flow rate can be changed onlywithin certain ranges, the refrigerant composition is themost flexible and significant variable in the design ofMR systems.

4. New Method for Selecting Mixed-RefrigerantComposition

The major difficulty in formulating the problem forthe selection of refrigerant compositions comes from thehighly interactive relationship between the variablesand the tight composite curves. Any change in therefrigerant composition, condensing/evaporating pres-sures, or refrigerant flow rate will alter the shape andposition of the hot and cold composite curves. Thus,changing the refrigerant flow rate will also change thehorizontal length of the two composite curves. Conse-quently, any small change in the variables might bemore than enough to violate the desired features andeven invalidate the heat transfer. A pure mathematicalprogramming approach is opaque to the users, who needto have a thorough understanding of and confidence inthe procedures and solutions. Because of the highlynonlinear nature of the problem, optimization can easilybe halted at infeasible points or trapped at local optima.

Without sufficient insight and understanding, it isdifficult to make further improvements.

In this paper, we propose a strategy for selectingmixed-refrigerant compositions by a combined NLP/thermodynamic approach. By using thermodynamics,the complex design problem can be displayed visually,and the interactions between variables can be lumpedtogether as the hot and cold composite curves. The NLPoptimizes the design variables to achieve the optimalsolution for a given objective function. “Perfect” match-ing between the hot and cold composite curves wouldindicate that the two lines are exactly parallel to eachother. The strategy is to find the refrigerant compositionwithin the given refrigerant set (say, C1-C4 andnitrogen) for which the hot and cold composite curvesare most parallel to each other. The overall procedureof the strategy is depicted in Figure 13.

The method starts with an initial setting of therefrigerant composition, flow rate, and evaporation andcondensation pressures. First, the hot composite curvefor the given operating conditions is generated. Then,we shift the hot composite curve down by a certaintemperature difference, say 5 °C, to generate the“pseudo-cold” composite curve. The pseudo-cold com-posite curve serves as a “target” for the optimization toachieve by changing the composition. An NLP model isused to perform an optimization to find the bestcomposition that can achieve the minimum value of theobjective function. The whole spectrum of the compositecurve is sliced into N intervals. The value of N ispredetermined by the user considering the tradeoffbetween the accuracy of the temperature profiles andthe required computation time. Within each interval i,the material and energy balances are used to calculatethe temperature of the pseudo-cold composite, Th i, andof the real cold composite, T̂i. The optimization algo-rithm continuously updates the composition of therefrigerants. When it agrees with the optimality criteriaof the optimization, the method determines the bestrefrigerant composition.

An NLP model is used to optimize the composition ofthe refrigerant mixture to target desired properties. Ageneric form for such a problem is

where ê′ represents the mole fraction variables and êrepresents the continuous variables.

In the above, ê is a vector of all continuous variables,and ê′ is a subset of ê that represents the mole fractionof each component in a refrigerant mixture. êu and êl

are specified upper and lower bounds, respectively. Forê′, the upper bound is 1 and the lower bound, 0. Inaddition to ê′, ê includes all thermodynamic and physi-cal properties of interest, such as the bubble- and dew-point temperatures of the refrigerant mixture, the latentheats of vaporization and condensation of a refrigerantmixture, and the specific heat of the vapor-phasemixture.

Figure 12. Effect of optimizing refrigerant compositions.

Problem NLP

minimize f(ê) (4)

subject to

g(ê) g 0

h(ê) ) 0

êl e ê e êu

ê′ ⊂ ê


Objective Function. There are three different formsof objective function:

(1) Minimization of the crossover: Minimize the singlelargest ∆Tmin violation, as shown in Figure 14. Theobjective function can be expressed as

This approach implies minimization of the single largesttemperature violation. The temperature constraint guar-antees that the cold composite curve is below the hotcomposite everywhere, so that heat transfer is alwaysvalid.

Because we consider only the single largest ∆Tminviolation, we can ignore the changes in other ∆Tminviolations.

(2) Minimization of the sum of the crossovers: Mini-mize the sum of the overall ∆Tmin violation, as shownin Figure 15. The objective function can be written as

If (Th i - T̂i) e 0, the extent of temperature violation hasno effect on the objective function. Compared withminimization of the single largest temperature violation,this objective function makes more thorough consider-ation of the entire shape of the hot and cold compositecurves.

For the same temperature shift for the pseudo-coldcomposite curve, this objective function often leads to ahigher shaftwork requirement than using minimizationof the crossover. The reason is that, although the resultsfrom minimization of the sum of the crossovers agreemore closely with the pseudo-cold composite curve, theresults from minimization of the crossover have overallsmaller temperature driving force between the hot andtrue cold composite curves. Nevertheless, this result canbe improved by decreasing the degree of temperatureshifting for the pseudo-cold composite curve.

Figure 13. Strategy for optimal selection of refrigerant compositions.

Figure 14. Illustration of using minimization of the crossover asthe objective function.

minimize y (5)

subject to

y g 0

y g Th i - T̂i g - y

T̂i e Th i + δ

i ∈ N

Figure 15. Illustration of using minimization of the sum of thecrossovers as the objective function.

minimize ∑i)1

N

yi (6)

subject to

y g 0

y g Th i - T̂i

T̂i < Th i + δ

i ∈ N


(3) Minimization of the shaftwork requirement: Usingthis objective function can often result in the lowestshaftwork requirement among the three types of objec-tive functions and seems the most straightforward wayof defining the objective function

Because this objective function ignores the shape of thecomposite curves, it sometimes yields a refrigerantcomposition that would cause water to be present in theinlet stream of the compressor. This objective functionworks best when the refrigerant flow rate and condens-ing/evaporating pressure ratio have been significantlyreduced, that is, approaching the end of the optimizationtask. Under such conditions, this optimization is freefrom the problem of water in the inlet stream and canusually give the lowest shaftwork requirement.

Equality Constraints. The equality constraints inthis problem formulation are mainly the equations forbalance and the calculation of physical properties. Thebalance equations are performed around a differentiatedsegment of the heat exchanger, as shown in Figure 16.The equality constraints are as follows:

The last term on the right-hand side of eq 10 is theenergy influx at segment i. The refrigerant mixture atsegment i absorbs an energy influx of this amount andevaporates part of liquid, Li - Li+1, to satisfy the energybalance.

The Peng-Robinson EOS is used to calculate thefugacity coefficients for both the vapor and liquidphases. An iterative computation is needed for thisconstraint. However, this constraint can be implicitlyincluded in the calculation of the evaporation temper-ature of a refrigerant mixture, because an equilibriumstate has to be reached to find the evaporation temper-ature and the corresponding vapor-phase composition.

This constraint ensures that the mole fractions of allcomponents that make up the mixture add up to 1 inboth the vapor and liquid phases.

The bubble-point temperature and vapor compositionare calculated for a liquid of known composition atpressure P. An initial guess is needed for the bubble-point temperature and the vapor mole fractions, andthen eq 12 and the equality of the component fugacities,eq 11, are checked for each component with the fugaci-ties calculated by an equation of state. Details of thecalculation can be found in Sandler.16

It is worth emphasizing that T̃i, the local temperatureof the hot composite curve, is calculated each time theNLP model is initiated or the composition is updatedby the optimizer. As a result, the hot composite curveand the pseudo-cold composite curve are not fixed in thismethod. The method for generating the hot compositecurve can be found in Linnhoff et al.17

Figure 16. System boundary for balance equations.

minimize ∑j)1

M

WSj (7)

subject to

T̂i < Th i + δ i ∈ N

mass and composition balance

Li + Vi ) Li+1 + Vi+1 (8)

Xj,iLi + Yj,iVi ) Xj,i+1Li+1 + Yj,i+1Vi+1

∀ i ∈ {1, ..., N}, ∀ j ∈ {1, ..., J} (9)

energy balance

LihiL + Vihi

V )

Li+1hi+1L + Vi+1hi+1

V + ∑k

CPk(Tk,ih - Tk,j+1

h )

∀ i ∈ {1, ..., N}, ∀ k ∈ {1, ..., K} (10)

phase equilibrium

Yj,iφjV(p,T̂,Yh ) ) Xj,iφj

L(p,T̂,Xh )∀ i ∈ {1, ..., N}, ∀ j ∈ {1, ..., J} (11)

sum of mole fractions

∑j

Yj,i ) 1 ∀ i ∈ {1, ..., N}, ∀ j ∈ {1, ..., J} (12)

∑j

Xj,i ) 1 ∀ i ∈ {1, ..., N}, ∀ j ∈ {1, ..., J} (13)

mixture evaporation temperature, T̂

T̂ ) T(P, Xh ) (14)

generation of Th

Ti ) T̃i - δ ∀ i ∈ {1, ..., N} (15)


By using the Peng-Robinson EOS, the enthalpy of thevapor and liquid phases of a refrigerant mixture can becalculated in a departure function form with referenceto ideal gas mixture

The subscript m indicates a mixture property, and thesuperscript IGM means a property of an ideal-gasmixture. Changing the subscript from V to L, eq 18 alsoapplied to liquid-phase mixtures.

The above equation states that the after-throttlingrefrigerant temperature is a function of the tempera-ture, pressure, and composition of the throttle inletliquid and the throttle outlet pressure, subject toisenthalpic changes.

Although no frictional pressure drop is assumed withinheat exchanger in this study, the constraint can bechanged to accommodate the condition once the pressureprofile is known inside the heat exchanger by using theform

where σ is the pressure drop between each interval.Inequality Constraints. The inequality constraints

are of two types: mole fraction constraints for eachcomponent in the refrigerant mixture and the temper-ature approach constraints within the heat exchanger.They are as follows

This constraint states that the temperature of the coldcomposite curve should be colder than that of the hotcomposite curve everywhere. In this way, heat-transfervalidity within a heat exchanger is enforced.

No matter which type of objective function is used forcomposition optimization, we should always check the

hot and cold composite curves to determine whether theresults are viable.

5. Systematic Synthesis of MR Systems

Judging from the success of the proposed method forthe optimal selection of refrigerant compositions, we candevelop a systematic method for the synthesis of MRsystems. Figure 17 explains the methodology. Theprocedure commences from an initial setting of therefrigerant flow rate, composition, and condensation andevaporation pressure levels. The initial guess of therefrigerant composition can be arbitrary, although theinitial condition does affect the optimization results. Asfor the initial refrigerant flow rate and pressure levels,we initially choose generous values, so as to leave roomfor the optimization to reduce their values.

For the design task of the PRICO process for LNG,reasonable ranges are 3-4 times the NG flow rate forthe refrigerant flow rate, 40-50 bar for the condensingpressure, and 4-5 bar for the evaporation pressure. Ifthe optimal selection of the refrigerant composition issuccessful for a given refrigerant flow rate and pressurelevels (evaporating and condensing), the refrigerant flowrate and/or pressure levels are adjusted or reduced, andthe procedure returns to the composition selection stage.After several iterations, the procedure terminates wheneither the refrigerant flow rate is too small or thepressure levels of condensing and evaporating are tooclose so that temperature crosses always occur in theheat exchanger. Therefore, no further improvement ispossible by changing the refrigerant composition, andthe optimization reaches the final design of the MRsystem by picking the best solution. Modifying therefrigerant flow rate or pressure levels can be doneusing heuristics, judgment, or optimization. The choiceof different objective functions, as will be illustrated inthe case studies later, can affect the final results.Nevertheless, the best solution to select is debatable.Very possibly, the solution that gives the lowest shaft-work requirement might incur an extra large heat-transfer area and increased capital costs. Pua et al.18

physical property constraints

hiL ) hL(P,T̂,Xh ) ∀ i ∈ {1, ..., N} (16)

hiV ) hV(P,T̂,Yh ) ∀ i ∈ {1, ..., N} (17)

hV(P,T̂,Yh ) - hIGM(P,T̂,Yh ) ) RT(Zm - 1) +

T(dam

dT ) - am

2x2bm

ln[Zm + (1 + x2)Bm

Zm + (1 - x2)Bm] (18)

after-throttling and after-mixing refrigeranttemperature

Tto ) f(Tti, Pti, Pto, Xh ti) (19)

subject to

hto ) hti

pressure profile constraint

Pi ) Pi+1 ∀ i ∈ {1, ..., N} (20)

Pi ) Pi+1 + σ ∀ i ∈ {1, ..., N} (21)

mole fraction constraint

0 e Yj,i e 1, 0 e Xj,i e 1∀ i ∈ {1, ..., N}, ∀ j ∈ {1, ..., J} (22)

temperature approach constraint

T̂i < Th i + δ ∀ i ∈ {1, ..., N} (23)

Figure 17. Proposed synthesis strategy for MR systems.


discussed this issue and combined the method presentedin this paper with a design method for the synthesis ofplate-and-fin heat exchanger networks. The resultsshow a tradeoff between shaftwork savings and heatexchanger network costs and provide better guidelinesfor selecting the most economic solution to minimize thetotal cost.

As for all NLP problems, the solutions are sensitiveto the initial conditions. However, it is interesting tonote that a feasible solution is not required as the initialstarting point to carry out the optimization. The use ofthe pseudo-cold composite curve helps to correct infea-sible intermediate solutions toward feasible ones. Afterthe first run of the optimization, a feasible set ofrefrigerant compositions usually will be found for aspecified refrigerant flow rate and pressure ratio.

This complete design method has been integrated andimplemented in STAR at the Department of ProcessIntegration, UMIST. A snapshot of the software inter-face is shown in Figure 18. All case studies presentedin this paper were run on STAR.

The method is readily extended to the design of morecomplex MR systems. Figure 19 shows a four-stage MR

system, which splits the expansion of liquid refrigerantinto four stages and still has only one compression train.

No matter how complex the system is, there will beonly one hot composite curve and one cold compositecurve. Therefore, by considering the match between thehot and cold composite curves, this method can bereadily applied to the synthesis of more complex MRsystems by changing the modeling of the NLP, but moredegrees of freedom (flow rates and pressures of eachsplitting) must be optimized.

6. Case Studies

Three case studies are used here to demonstrate thenew synthesis method of MR systems by using thePRICO process as our base case, as illustrated in Figure20.

The natural gas enters the heat exchanger at ambienttemperature and high pressure and is to be liquefiedby the mixed refrigerant flowing countercurrentlythrough the heat exchanger. The refrigerant then passesthe compressor to recover its pressure back to thecondensing condition. After the partial condenser, nor-

Figure 18. Snapshot of the user interface in STAR for the design of complex MR systems.

Figure 19. Four-stage MR system.


mally cooled by cooling water, the refrigerant is onlypartially condensed. Because there is no other heat sinkin the process that can totally condense the refrigerant,it must be condensed by the cold refrigerant itself.

Case study 1 is to design the PRICO process toachieve the lowest shaftwork consumption by the newmethod. Minimization of the sum of the crossovers isemployed as the default objective function. In case study2, different objective functions are used, with theexpectation that further energy savings can be achieved.Shifting of the hot composite curve by a chosen tem-perature difference forms the pseudo-cold compositecurve. Certainly, the degree of shifting should have asignificant effect on the final optimal solutions. In casestudy 3, different temperature shifts are tried to see howthe optimization results are influenced.

Table 1 lists the properties of the segmented naturalgas feed stream. We can see that in the temperaturerange between -70.10 and -82.26 °C, the CP of the feedstream is exceptionally high, which implies a majorliquefaction process has happened.

Case Study 1. Minimization of the sum of thecrossovers is used as the objective function. The opti-mization steps are presented in Table 2.

The procedure starts with an initial setting of thepressure levels, refrigerant flow rate, and composition.

Figure 20. Design task for a PRICO process.

Figure 21. Composite curves for the final design.

Table 1. Segmented Natural Gas Feed Stream

stream Ts (°C) Tt (°C) DH (kW) CP (kW/°C)

1.1 25.00 -6.03 -1861.5 60.01.2 -6.03 -34.09 -1964.3 70.01.3 -34.09 -57.65 -1885.0 80.01.4 -57.65 -70.10 -2490.0 200.01.5 -70.10 -74.55 -1780.0 400.01.6 -74.55 -82.26 -3084.0 400.01.7 -82.26 -96.50 -1424.0 100.01.8 -96.50 -115.00 -1850.0 100.01.9 -115.00 -163.00 -3840.0 80.0

Table 2. Optimization Steps

pressurelevels(bar)

composition(wt %)

refrigerantflow rate(kmol/s)

shaftwork(kW)

initial 4.0/46.0 C1, 28.9; C2, 37.5 4.0 Xa

C3, 16.5; C4, 4.8N2, 12.3

1 4.0/46.0 C1, 15.7; C2, 46.4 4.0 34 852.3C3, 3.5; C4, 14.7N2, 19.8

2 4.0/45.0 C1, 18.7; C2, 45.1 3.8 32 610.6C3, 0.0; C4, 18.1N2, 18.2

3 3.8/43.0 C1, 20.1; C2, 42.44 3.6 30 912.6C3, 4.5; C4, 17.1N2, 15.9

4 3.8/42.0 C1, 24.1; C2, 39.8 3.4 28 763.9C3, 0.36; C4, 21.5N2, 14.3

5 3.7/40.0 C1, 18.5; C2, 47.0 3.3 27 5915C3, 0.01; C4, 20.6N2, 13.9

6 3.7/40.0 C1, 21.7; C2, 40.8 3.1 Xa

C3, 14.6; C4, 13.3N2, 9.6

a Temperature cross.


When we check the hot and cold composite curves ofthe initial setting, a temperature cross occurs inside theheat exchanger. The following steps keep changing thepressure levels (so that the pressure ratio decreases) andreducing the refrigerant flow rate and use the optimalselection of the refrigerant composition to find the bestcomposition under each condition. The procedure stopsat the 6th step when temperature crosses always occurand no further improvement is possible. The final designof the PRICO process is obtained from the 5th stage,the last successful run in the procedure for the optimalselection of the refrigerant composition. Figure 21 showsthe hot and cold composite curves of our final design. Ifwe compare the results with those of the commercialPRICO process, we achieve an energy savings of 21.3%.

Case Study 2. In this paper, three different forms ofobjective function have been proposed, and each has itsown characteristics and strengths. Minimization of theshaftwork consumption usually generates the design ofthe lowest shaftwork consumption, but it can also resultin water in the inlet stream to the compressor when therefrigerant flow rate is high. Because the refrigerantflow rate in the previous case study has been reducedsignificantly by the 5th stage, it is safe to try usingminimization of the shaftwork requirement as theobjective function to achieve further shaftwork savings.As shown in Table 3, a further 3.6% savings in shaft-work requirement is achieved at the {6′}th stage byswitching the objective function form to minimizationof the shaftwork requirement from the (5′)th step.

This demonstrates that different objective functionsperform well under different conditions. The generalguidelines are as follows: When the refrigerant flowrate is still high, minimization of the crossover andminimization of the sum of the crossovers give satisfac-tory and reliable results. Minimization of the shaftworkrequirement reaches greater savings when the refriger-ant flow rate is reasonably low, thus decreasing thepossibility that the compressor inlet stream containswater. Using minimization of the crossover or minimi-zation of the sum of the crossovers as the objectivefunction first is suggested, and when this optimizationterminates, then one should switch to minimization ofthe shaftwork to explore further possibilities for shaft-work savings. Table 4 provides a comparison with theresults for the commercial PRICO process as reportedin Finn et al.15

Case Study 3. The degree of temperature shiftingin the previous two case studies is 5 °C. In this casestudy, different degrees of temperature shifting are triedto see the impact on the final optimization results. Tomaintain consistency, minimization of the sum of the

crossovers is used as the objective function throughoutcase study 3. Table 5 lists the results obtained by usingdifferent degrees of temperature shifting.

The optimal shaftwork consumption is progressivelyincreased with increasing degree of temperature shift-ing. This is in accordance with the fact that a largerdegree of temperature shifting for the pseudo-coldcomposite curve should generate a design with a widergap between the hot and cold composite curves, thusincreasing the energy loss within the heat exchangers.In other words, a higher shaftwork consumption isrequired. This adds another dimension of complexity offinding the truly optimal tradeoff between energy sav-ings and capital costs.

7. Conclusions

The difficulty in designing MR systems mainly stemsfrom two aspects: one is the expensive and highlynonlinear nature of computation, and the other is thesensitivity of the systems to operating changes, espe-cially changes in the composition of the refrigerantmixture. Few successful synthesis methods for MRsystems have been developed. In this paper, a novelmethod for the selection of refrigerant compositionsbased on a combined NLP/thermodynamic approach hasbeen proposed. Using this approach, a systematic syn-thesis tool for the complete design of MR systems hasbeen developed. This approach combines the power ofthermodynamics and mathematical programming. WhileNLP can satisfactorily provide the optimal choice ofprocess operating conditions, thermodynamics can, atthe same time, provide insights into and confidence inthe solution. Three key design variables, the condensa-tion and evaporation pressure levels, refrigerant flowrate, and refrigerant composition, are also discussed.Refrigerant composition is the most flexible and signifi-cant variable among the three. A case study, using thePRICO process as the base case, demonstrates that upto 25% savings in the shaftwork requirement comparedto that of the commercial PRICO process can be achievedby using this method.

Although the solution procedures of this methodpresented in this paper are not completely automatic,automation of the method using several successfulMINLP formulations (e.g., Duran and Grossmann19) ispossible and will be explored in a future study. Potential

Table 3. Effect of Changing Objective Function afterStep 5

pressurelevels(bar)

composition(wt %)

refrigerantflow rate(kmol/s)

shaftwork(kW)

5′ 3.7/4.0 C1, 25.9; C2, 36.4 3.3 26 679.9C3, 4.49; C4, 22.1N2, 11.2

6′ 3.7/40.0 C1, 27.3; C2, 35.6 3.2 26 601.5C3, 5.20; C4, 20.9N2, 11.0

7′ 3.6/40.0 C1, 25.8; C2, 36.6 3.1 Xa

C3, 0.15; C4, 15.9N2, 21.6

a Temperature cross.

Table 4. Comparison of Commercial PRICO Process andTwo Design Results

commercialPRICO

case study1a

case study2b

refrigerant composition - C1-C4, N2 C1-C4, N2shaftwork

(kJ/kg of LNG)1485.0 1168.6 1126.7

a PRICO by optimal selection of composition. b PRICO by opti-mal selection of composition using different objective functions.

Table 5. Results Obtained by Using Different Degrees ofTemperature Shifting (Case Study 3)

temperatureshift (°C)

shaftwork(kW)

3 26 857.94 27 243.55 27 591.56 29 323.57 30 142.88 31 115.6


applications of this method are in the design of morecomplex MR systems to fit various industrial needs andonline control strategies.

Notation

Parameters and Variables

B ) parameter in the Peng-Robinson EOSC ) number of components in the Gibbs phase ruleCP ) specific heatDH ) dutyF ) degrees of freedom in the Gibbs phase rulehIGM ) enthalpy of an ideal gas mixturehi

L ) enthalpy of the liquid phase in the ith intervalhi

V ) enthalpy of the vapor phase in the ith intervalhti ) enthalpy of the fluid at the inlet of the throttling valvehto ) enthalpy of the fluid at the outlet of the throttling

valveLi ) liquid mass flow rate in interval iJ ) number of componentsM ) number of independent reactions in the Gibbs phase

ruleN ) number of intervalsP ) pressureQ ) number of phases in the Gibbs phase ruleR ) universal gas constantT ) temperatureTh i ) temperature of interval i on the pseudo-cold composite

curveT̂i ) temperature of interval i on the real cold composite

curveTj,i

h ) temperature of the jth hot stream in interval iTs ) supply temperatureTt ) target temperatureTti ) temperature of a fluid at the inlet of the throttling

valveTto ) temperature of a fluid at the outlet of the throttling

valveVi ) vapor mass flow rate in interval iWSj ) shaftwork requirement of compressor jZ ) compressibility factorXj,i ) mole fraction of component j in interval i in the liquid

phaseYj,i ) mole fraction of component j in interval i in the vapor

phasex ) liquid-phase composition vectory ) vapor-phase composition vector

Greek Letters

φi ) fugacity coefficient of component iê ) vector of all continuous variablesê′ ) subset of ê, comprising the mole fraction of each

componentêu ) upper bound of vector ê

êl ) lower bound of vector êδ ) specified minimum temperature approach

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Received for review January 22, 2002Revised manuscript received June 27, 2002

Accepted June 27, 2002

IE020057P


optimal synthesis of mixed-refrigerant systems for low-temperature processes

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