jørgen bauck jensen optimal operation of refrigera- tion

Joslashrgen Bauck Jensen

Optimal Operation of Refrigera-tion Cycles

Doctoral thesisfor the degree of philosophiae doctor

Trondheim May 2008

Norwegian University of Science and TechnologyFaculty of Natural Sciences and TechnologyDepartment of chemical engineering

NTNU

Norwegian University of Science and Technology


Faculty of Natural Sciences and TechnologyDepartment of chemical engineering

ccopy 2008 Joslashrgen Bauck Jensen

ISBN 978-82-471-7801-0 (printed version)ISBN 978-82-471-7815-7 (electronic version)ISSN 1503-8181

Doctoral theses at NTNU 200891

Printed by tapir uttrykk

iii

Abstract

The theme of this thesis is optimal operation of refrigeration cycles but the resultsalso applies to the reverse process generating heat

A simple refrigeration cycle has five steady-state degrees of freedom related tocompressor choke valve heat transfer in condenser heat transfer in evaporatorand ldquoactive chargerdquo One degree of freedom is used to control the load of thecycle eg the compressor It is usually optimal to maximize the heat transfer inthe condenser and in the evaporator (eg by maximizing the fan speeds) The tworemaining degrees of freedom may be used to control the degree of super-heatingand the degree of sub-cooling It is found that super-heating should be minimizedwhereas some sub-cooling is optimal in terms of cycle efficiency The degree offreedom related to active charge is often lost by specifying no sub-cooling by thedesign This gives a loss in the order of 2 Different designs for affecting theactive charge are discussed

By allowing for sub-cooling in the condenser of a sub-critical refrigerant cyclethere are no fundamental differences between a trans-critical cycle (eg withCO2

as working fluid) and a sub-critical cycle (eg with ammonia as working fluid)However the practical operation is quite different For a sub-critical ammoniacycle several simple control structures gives close to optimal operation For thetrans-criticalCO2 cycle on the other hand a combination of measurements is nec-essary

Refrigeration processes are often designed by specifying a minimum approachtemperature (∆Tmin) in the evaporator and in the condenser With this approachthe optimality of sub-cooling in the condenser will not be found In additionspecifying the areas found by designing with the∆Tmin-method and re-optimizingwithout the constraints on∆Tmin leads to a different operating point These twodeficiencies shows that the∆Tmin-method is not sufficient An alternative method(simplified TAC) is proposed and compared with the∆Tmin-method

Considering the large amount of work that goes into the design of LNG processesthere is surprisingly little attention to their subsequent operation This partly comesfrom the assumption that optimal design and optimal operation is the same but thisis generally not true

The PRICO LNG process is used as an example for optimal design and optimaloperation of LNG processes In the design phase there are a number ofconstraintsthat must be satisfied to get a feasible design The limitations imposed by theconstraints are discussed It is found that constraints related to the compressor per-

iv

formance is important to consider In operation the objective function is muchsim-pler than in design where also the equipment is part of the optimization Two mainmodes of operation are studied i) minimize compressor shaft work for given pro-duction and ii) maximum production (eg for maximum compressor shaft work)

An important issue for optimal operation and plantwide control is to find the de-grees of freedom available for optimization A previously published systematicapproach to determine the steady-state degrees of freedom is extended totake intoaccount the active charge and the refrigerant composition as possible degrees offreedom A number of case studies are used to illustrate the findings

v

Acknowledgement

Most of all I would like to thank my supervisor Sigurd Skogestad for givingmethe chance to work on this thesis His support and guidance has helped me throughthese four years I would also like to thank people at Statoil and Linde for arrangingmy stay and work at the offices of Linde Engineering in Munich The key personsresponsible for providing this opportunity are Jostein Pettersen and HeinzBauer

The Process Systems Engineering Group has been may daily colleagues and Iwould like to thank the people that has been working there for a nice workingenvironment and a lot of interesting discussions

Finally I would like to thank my girlfriend Christina for her support and under-standing


vi

Contents

List of Figures xi

List of Tables xv

1 Introduction 111 Motivation 112 Thesis overview 213 Publication list 3

2 Introduction to vapour compression cycles 521 Introduction 5

211 Fundamentals 6212 Expansion device 8213 Condenser 9214 Evaporator 10215 Active charge 11216 Operation of simple cycles 12

22 Some design features 13221 Constant versus varying temperature loads 13222 Two-stage expansion and two pressure levels 14223 Internal heat exchange 16224 Cascaded cycles 16

23 Conclusion 17Bibliography 17

3 Simple cycles Part I 1931 Introduction 2032 Degrees of freedom in simple cycles 23

321 Design versus operation 23322 Active charge and holdup tanks 23

vii

viii Contents

323 Degrees of freedom for operation 2733 Discussion of some designs 28

331 Optimal designs 28332 Non-optimal designs 29

34 Optimality of sub-cooling 31341 Ammonia case study 31342 Explanation 34343 Discussion of sub-cooling Why not found before 35

35 Discussion 37351 Super-heating by internal heat exchange 37352 Selection of controlled variable 38


4 Simple cycles Part II 4141 Introduction 4142 Selection of controlled variable 43

421 Linear analysis 44422 Combination of measurements 45

43 Ammonia case study 46431 Modelling 46432 Optimal steady-state operation 47433 Selection of controlled variables 47

44 CO2 case study 53441 Modelling 54442 Optimal operation 54443 Selection of controlled variable 55

45 Discussion 60451 Super-heating 60452 Heat transfer coefficients 61453 Pressure control 61


5 Problems with the minimum approach temperature 6551 Introduction 6552 Motivating example Ammonia refrigeration cycle 68

521 ∆Tmin design-method 6953 Proposed simplified TAC method 71

531 Revisit of ammonia case study 7354 Other case studies 73

Contents ix

541 CO2 air-conditioner 73542 PRICO LNG process 75

55 Discussion 76551 ∆Tmin-method 76552 Other approaches 77553 Heat exchanger network design 77


6 Optimal design of a simple LNG process 8161 Introduction 81

611 Optimal design 83612 Design constraints 85

62 Process description 8663 Results for optimal design 88

631 An alternative design optimization 9464 Discussion 95

641 Compressor 95642 Heavy extraction 96643 Feed pressure 97644 Large PRICO plants 97


7 Optimal operation of a simple LNG process 10171 Introduction 10172 Process description 102

721 Nominal conditions 103722 Manipulated inputs 104723 Operational constraints 105

73 Model 106731 Compressor characteristic 106

74 Objective function 11075 Nominal optimum given production case (Mode I) 11176 Nominal optimum maximum production case (Mode II) 11277 Optimum with disturbances 114

771 Selection of controlled variables 11678 Discussion 120

781 Moving temperature profile 120782 Compressor characteristic 121783 Refrigerant composition 121

x Contents

784 Additional considerations 121Bibliography 121

8 Degrees of freedom for refrigeration cycles 12581 Introduction 12582 Degrees of freedom 126

821 Remark on active charge or ldquofeedrdquo for closed cycles 129822 Actual degrees of freedom for refrigerant cycles 130

83 Case studies 133831 Two-pressure level refrigeration 133832 Heat integrated distillation 134833 Small scale LNG process 137834 Propane pre-cooled mixed refrigerant (C3MR) 138835 Mixed fluid cascade (MFC) 141

84 Discussion 143841 Degrees of freedom 143842 Refrigerant composition 144843 Saturation in condenser 144


9 Conclusion 147

A Optimal operation of a mixed fluid cascade LNG plant 149A1 Introduction 149A2 Process description 150A3 Degree of freedom analysis 151

A31 Manipulated variables (MVrsquos) 152A32 Constraints during operation 152A33 Active constraints 152A34 Unconstrained degrees of freedom 153

A4 Optimization results 153A5 Control structure design 155A6 Conclusion 156Bibliography 157

B Trade-off between Energy Consumption and Food Quality Loss 159B1 Introduction 160B2 Process description 160

B21 Degree of freedom analysis 161B22 Mathematical model 162

Contents xi

B23 Influence of setpoints on energy consumption 162B24 Influence of setpoint on food quality 162

B3 Problem formulation 164B4 Optimization 166

B41 Optimization 166B42 Optimization results 166B43 Trade-off between energy consumption and food quality loss167

B5 Discussion 168B6 Conclusion 168Bibliography 169

xii Contents

List of Figures

21 A simple refrigeration cycle with pressure-enthalpy diagram 622 Schematic illustration of a Carnot 723 Expansion paths for turbine valve and a combination 824 Two basic types of condensers 925 Two basic types of evaporators 1026 A simple design with external fillingemptying system 1127 A possible control configuration for a simple refrigerant cycle 1328 Evaporator temperature profile 1429 Two stage expansion improves cycle efficiency 14210 Cooling at two pressure levels 15211 Internal heat exchange 16212 Cascaded cycles shown with internal heat exchange 17

31 A simple refrigeration cycle with pressure-enthalpy diagram 2032 Simple cycle with variable active charge 2533 Condenser with saturation at outlet (common but non-optimal) 2734 Evaporator with saturation at outlet (optimal) 2735 Two potentially optimal designs 2936 Three non-optimal designs 3037 Cold warehouse with ammonia refrigeration unit 3138 Pressure-enthalpy diagrams with and without sub-cooling 3239 Temperature profile in condenser 33310 Pressure-enthalpy diagram for a cycle with and without sub-cooling34311 Pressure-enthalpy diagram for infinite area case 36312 Internal heat exchange 38

41 Simple refrigeration cycle studied in this paper 4242 Optimal operation for the ammonia case study 4743 Ammonia case Compressor power and loss for disturbances 5044 Ammonia case Loss as function of implementation error 52

xiii

xiv List of Figures

45 Proposed control structure for the ammonia cycle 5346 TheCO2 cycle is trans-critical and has an internal heat exchanger 5447 CO2 case Temperature profiles 5648 CO2 case Compressor power and loss for different disturbances 5849 CO2 case Loss as function of implementation error 59410 Proposed control structure for theCO2 cycle 60411 Alternative cycle with liquid receiver on high pressure side 61

51 The effect of different values for∆Tmin 6652 An ammonia refrigeration system 6853 Temperature profile in the condenser for the∆Tmin-method 70

61 Simplified flowsheet of the PRICO process 8262 Further simplified flowsheet of the PRICO process used in this work8763 Illustrating the use of liquid turbine and valve as expansion device9364 Alternative locations of the NGL recovery 9665 Maximum LNG production as function of feed pressure 97

71 A simplified flowsheet of the PRICO process 10372 A cubic compressor characteristic curve 10873 Compressor map for the refrigerant compressor 10974 Nominal P-h diagram for mode II 11275 Nominal compressor operating point 11376 Temperature profiles for the maximum production case 11477 Shaft work as function of disturbances for mode I 11778 LNG production as function of disturbances for mode II 11879 ∆Tsup as function of disturbance inWmax

s 119710 LNG production as function ofN 120

81 A simple refrigeration cycle with 5 potential degrees of freedom 12882 A simple (not closed) refrigeration cycle 13083 Simple cycle with liquid receiver on the high pressure side 13184 Cooling at two pressure levelsNss= 4 13485 Two designs of heat integrated distillation 13586 A small scale LNG concept 13787 Flowsheet of the C3MR process 13988 Flowsheet of the MFC process 142

A1 Simplified flowsheet of the MFC process 151A2 Temperature profiles 154A3 Suggested control structure for the MFC process 155

List of Figures xv

B1 Simplified supermarket refrigeration system 161B2 Energy consumption under different setpoints 164B3 Fresh fish quality loss when stored at different temperatures 164B4 Optimization between food quality loss and energy consumption 168B5 Traditional operation (Case 1) 169B6 Optimal operation forTcabin= 1C (Case 2) 170B7 Optimal operation forT food = 1C (Case 3) 171B8 Optimal operation forQfood le 755 (Case 4) 172

xvi List of Figures

List of Tables

31 Structure of model equations 2132 Typical specifications in design and operation 2333 Optimal operation with and without sub-cooling 33

41 Structure of model equations 4642 Data for the ammonia case study 4743 Optimal steady-state for ammonia case study 4844 Linear ldquomaximum gainrdquo analysis for the ammonia case 4945 Conditions for theCO2 case study 5546 Optimal operation forCO2 case 5547 Linear ldquomaximum gainrdquo analysis for theCO2 case 57

51 Structure of model equations for the ammonia case study 6952 Ammonia case study 7153 CO2 air-conditioner 7454 PRICO LNG process 76

61 Maximum head per compressor wheel 8662 Design constraints based on data from Price and Mortko (1996) 8963 Optimal design results for nine different cases 89

64 Optimal design withC0 = 110middot103kgs-1(m-2)065

95

71 The nominal operating point for mode I and II 11172 Nominal minimum and maximum values for the disturbances 11473 Structure of model equations 12374 Data for the PRICO process 12375 Optimal operation with disturbances 124

81 Potential operational degrees of freedom for typical process units 12782 Actual degrees of freedom for refrigeration cycles 129

xvii

xviii List of Tables

A1 Optimal operation of a MFC process 155

B1 Model equations 163B2 Some data used in the simulation 163B3 Traditional operation and optimal operation for different constraints166

Chapter 1

Introduction

11 Motivation

The demand for energy is increasing rapidly and it is expected to increaseevenfaster in the years to come At the same time the amount of oil is decreasing soalternative transportable energy sources has gained more attention Oneof thesealternatives is liquefied natural gas (LNG) which is natural gas (mainly methane) inliquid state at atmospheric pressure and aboutminus160C The process of liquefyingnatural gas requires large amounts of energy so a lot of work has beendone ondesign of LNG processes However it seems that the subsequent operation is lessstudied at least in the open literature This is a bit surprising considering that evensmall improvements will give large savings due to high throughput

An important issue for optimal operation is to find the steady-state degrees offree-dom available for optimization (Nopt) This number is important for several rea-sons First it determines the degrees of freedom available for solving theoptimiza-tion problem However more importantly in terms of operation it determines thenumber of steady-state controlled variables that need to be selected Optimalop-eration is normally implemented by keeping those variables at constant setpointsThe selection of controlled variables is therefor also an important issue The ob-jective is to achieve ldquoself-optimizingrdquo control where a constant setpoint for theselected variable indirectly leads to near-optimal operation Note that the selectionof a good controlled variable is equally important in an ldquoadvancedrdquo controlschemelike MPC which also is based on keeping the controlled variables close to givensetpoints

Our goal was to study optimal operation of refrigeration cycles used for liquefac-

1

2 Introduction

tion of LNG but as a start we needed to understand operation of simpler refrig-eration cycles We therefore started out by studying basic refrigerationcycles andmoved in the direction of refrigeration processes used for liquefaction ofnaturalgas

12 Thesis overview

We discuss degrees of freedom for simple refrigeration cycles in Chapter 3 Asimple ammonia refrigerator model is used for numerical results The main focusis to show that allowing for sub-cooling in the condenser gives one extra steady-state degree of freedom related to ldquoactive chargerdquo This shows that there are nofundamental differences between a sub-critical cycle and a trans-critical cycle (egtheCO2 cycle presented in Chapter4) We also show that some sub-cooling isbeneficial in terms of cycle efficiency

Chapter4 is dedicated to selection of controlled variables for simple refrigerationcycles We present two different case studies a conventional sub-critical ammoniacycle and a trans-criticalCO2 cycle We find that the ammonia cycle has severalcontrol structures that will give close to optimal operation For theCO2 cyclehowever a combination of controlled variables is necessary to give acceptableperformance

We address the problem of specifying the minimum approach temperature (∆Tmin)as a method of designing processes with heat exchangers in Chapter5 A simpledesign method for preliminary designs are presented and compared with the∆Tmin

method

In Chapter6 we discuss design optimization of a simple LNG process the PRICOprocess Nine cases are presented to show the effect of different constraints onthe optimum The PRICO process is studied further in Chapter7 where the themeis optimal operation and selection of controlled variables We present differentmodes of operation and find a self-optimizing control structure

Chapter8 discuss operational degrees of freedom for refrigeration cycles and sum-marize some of the important results for degree of freedom analysis Several illus-trative process examples are presented to illustrate our findings These cases areincluding heat integrated distillation and two complex processes for liquefactionof natural gas the mixed fluid cascade (MFC) process from Statoil-LindeLNGTechnology Alliance and the propane pre-cooled mixed refrigerant (C3MR) pro-cess from Air Products

13 Publication list 3

AppendixA presents some results from optimization on the MFC process Weassumed that the degree of super-heating was controlled at 10C and optimizedthe remaining 13 unconstrained degrees of freedom (including 9 compositions)The paper only presents the nominal operating point so it still remains to find aself-optimizing control structure

In AppendixB we consider a refrigerator that is used to refrigerate food cabinetinside a store Optimal operation of this refrigerator is studied with respect tobothfood quality and energy consumption by using a simple model for the food qualityand a simple model for the refrigerator performance We compare the traditionalway of controlling such refrigerators with different improved schemes The mainsavings are from letting the condenser pressure vary with the ambient temperatureHowever there are also some savings related to reducing the temperature inthefood cabinet during the night when the refrigerator has higher efficiency

13 Publication list

Journal papers

Chapter 3

J B Jensen and S Skogestad (2007) ldquoOptimal operation of simple refrigerationcycles Part I Degrees of freedom and optimality of sub-coolingrdquo Comput ChemEng 31 712-721

Chapter 4

J B Jensen and S Skogestad (2007) ldquoOptimal operation of simple refrigerationcycles Part II Selection of controlled variablesrdquo Comput Chem Eng 31 1590-1601

Chapter 5

J B Jensen and S Skogestad ldquoProblems with specifying∆Tmin in design of pro-cesses with heat exchangersrdquoAccepted for publication in Ind Eng Chem Re-search

4 Introduction

Chapter 8

J B Jensen and S Skogestad ldquoDegrees of freedom for refrigeration cyclesrdquoSub-mitted to Ind Eng Chem Research

Conferences

Chapter 4

J B Jensen and S Skogestad ldquoControl and optimal operation of simple heatpump cyclesrdquoEuropean Symposium on Computer Aided Process Engineering -15 Barcelona Spain 2005 Published by Elsevier ISBN 0-444-51987-4 p 1429-1434

Chapter 7

J B Jensen and S Skogestad ldquoOptimal operation of a simple LNG processrdquoInternational Symposium on Advanced Control of Chemical Processes GramadoBrazil 2006

Appendix A

J B Jensen and S Skogestad ldquoOptimal operation of a mixed fluid cascade LNGplantrdquo 16th European Symposium on Computer Aided Process Engineering (ES-CAPE) and 9th International Symposium on Process Systems Engineering (PSE)Garmisch-Partenkirchen Germany 2006 Published by Elsevier ISBN0-444-52969-1 p 1568-1574

Co-authored papers

Appendix B

J Cai and J B Jensen and S Skogestad and J Stoustrup ldquoBalanceEnergy Con-sumption and Food Quality Loss in Supermarket Refrigeration SystemrdquoSubmit-ted for publication in the proceedings of the 2008 American Control ConferenceSeattle USA

Chapter 2

Introduction to vapourcompression cycles

21 Introduction

In this chapter it is given an overview of some basic vapour compressioncycledesigns The goal is not to cover all designs but we wish to introduce thefeaturesused later in the thesis

The vapour compression cycle is the most common process used in both refriger-ation systems and heat pumps The two processes operate in the same manner Asimple flowsheet and pressure enthalpy diagram with the nomenclature is given inFigure21 For refrigeration it is the cooling dutyQC that is of interest whereas itis the heating dutyQH that is of interest for a heating cycle When a heat pump isused in heat integration both the heating and cooling duty is utilized

Some typical areas of usage are

Household Refrigerators air-conditioners and heat pumps

Automotive Air-conditioners and refrigerators

Industry Refrigeration of process streams heat pumps for heat integration

5

6 Introduction to vapour compression cycles

QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup

(b) Pressure enthalpy diagram

Figure 21 A simple refrigeration or heat pump cycle with a corresponding typicalpressure-enthalpy diagram indicating both sub-cooling and super-heating

211 Fundamentals

The basic refrigeration (or heating) cycle has four states (denoted 1234 in Figure21) and operates in the following manner

The working fluid is evaporated and possibly super-heated (3rarr 4) by heat ex-change with the cold source (eg air inside the refrigerator) Energy is addedin a compressor (eg as electricity) to increase the pressure of the working fluid(4 rarr 1) The high pressure vapour is de-super-heated condensed andpossiblysub-cooled (1rarr 2) by heat exchange with the hot source (eg air in the room)The liquid is then expanded through an expansion device (choke valve) (2rarr 3) togive a low temperature two-phase mixture at the evaporator inlet

The efficiency of a vapour compression cycle is often reported in terms ofldquocoef-ficient of performancerdquo (COP) The COP for for a heating and cooling process isgiven by

COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively

The vapour compression cycle for heating or cooling has some similarities withcyclic processes for generating mechanical work from heat eg steamturbine cy-cles These work generating cycles were studied extensively during the1800 and

21 Introduction 7

Nicolas Leonard Sadi Carnot studied the theoretical cycle later called the Carnotcycle He found that the maximum work that may be extracted from a given hotsource is only depending on the hot source temperatureTH and the cold sink tem-peratureTC This may be derived using the 2nd law of thermodynamics∆Stotalge 0where the equality holds for an ideal reversible process

QH

QC

WsIdeal cyclic machine

Cold sink (TC)

Hot source (TH)

Reversible entropy change

∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC

Figure 22 Schematic illustration of a Carnot machine generating mechanicalwork from a hot source

The Carnot machine is illustrated in Figure22 The cyclic machine will have nonet change in entropy so∆Smachine= 0 If we assume reversible heat transfer andthat the hot source and cold sink has constant temperature we get∆SH = minusQH

THand

∆SC = QCTC

Thus for the reversible case we get

QH

TH=

QC

TC(23)

Taking an energy balance around the machineW = QH minusQC we may express theabove as follows

W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH

= ηCarnot is named the Carnot efficiency This is the theoretical max-imum fraction of the heatQH that may be converted to mechanical workWs for an


ideal reversible process CorrespondinglyTHTC

minus1 is the minimum fraction of thecooling dutyQC that must be added as mechanical workWs Thus for a work gen-erating process a large difference betweenTC andTH is beneficial For a coolingprocess however the opposite is true

212 Expansion device

In household refrigerators a fixed expansion device called capillary tube is oftenused This is a thin long pipe that gives the necessary pressure drop from the con-denser to the evaporator The operational characteristics of systems withcapillarytubes is discussedDossat(2002 page 356)

For larger systems with more variations in operating conditions it is desirable tohave a variable expansion device The most common is a choke valve where thepressure is reduced without doing any work (isenthalpic expansion)

P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)

(a) Different expansion paths

Turbine for liquidexpansion

Valve with smallpressure drop

Tank withsaturation

Valve for two-phase expansion

(b) Liquid turbine and choke valve path (iii)

Figure 23 Expansion in turbine increase the system performance The left figureshows different expansion alternatives in a pressure enthalpy diagram i) chokevalve ii) turbine and iii) combination of liquid turbine and choke valve shown inright figure

The choke valve may be replaced by a turbine to improve the efficiency of thecy-cle The pressure enthalpy diagram in Figure23(a)illustrates the different pathsfor (i) isenthalpic expansion (choke valve) and (ii) isentropic expansion(ideal tur-bine) The vapour fraction (and specific enthalpy) into the evaporator islower

21 Introduction 9

with a turbine than for a choke valve so the COPc is higher becauseh4 minus h3 islarger without affectingh2minush1 A practical problem of utilizing a turbine is thatsome of the liquid becomes vapour during expansion and this may cause wearinthe turbine A solution to this problem is to use a combination of a liquid tur-bine and a choke valve illustrated in Figure23(b)(Barclay and Yang 2006) Theturbine will then expand the liquid down to a pressure slightly higher than the sat-uration pressure and the choke valve will handle the remaining pressure drop intothe two-phase region see path (iii) in Figure23(a) This is only useful if the liq-uid is sub-cooled before the expansion device otherwise there will be nopressuredrop for the liquid turbine

The inclusion of a liquid turbine is considered in Chapter6

213 Condenser

The design of the condenser is important for the possible extra degree offreedomrelated to active charge Two basic types of condensers are shown in Figure24Figure24(a)shows a case where the liquid drains into a tank below the heat trans-

To evaporator From evaporator

(a) Saturation at outlet


∆Tsub

Air fan

(b) Plug-flow with possible sub-cooling

Figure 24 Two basic types of condensers

fer zone With this design there will be no sub-cooling in the condenser sothereis no degree of freedom related to active charge since additional charge will onlychange the level in the tank below the condenser An alternative design would beif the liquid covers part of the heat transfer zone but this is not considered here

Figure24(b)shows a condenser with plug-flow The refrigerant is first desuper-heated then condensed and finally sub-cooled Here the degree of freedom related


to active charge may be available (if we have means of changing it) becausechang-ing the charge will directly affect the pressure and sub-cooling in the condenser

Remark The plug-flow condenser may also give no sub-cooling by adding a liquid re-ceiver after the condenser

214 Evaporator

The design of the evaporator influences the degree of super-heating before thecompressor No super-heating is achieved with the flooded evaporator shown inFigure25(a) Here a float controller may adjust the choke valve to maintain aconstant liquid level but other strategies are also possible Figure25(b)shows aplug flow evaporator where we may have super-heating at the outlet The super-heating may be controlled by a thermostatic expansion valve (TEV) The TEV willadjust its opening to give a certain degree of super-heating out of the evaporatorThe TEV requires a certain degree of super-heating to be able to measureit and toassure that no liquid is fed to the compressor also dynamically The super-heatingis therefor typically controlled at 10C Langley(2002 page 43)

From condenser To condenser



∆Tsup

Air fan

(b) Plug-flow with possible super-heating

Figure 25 Two basic types of evaporators

Remark The plug-flow evaporator may also give no super-heating by adding a liquidreceiver after the evaporator

21 Introduction 11

215 Active charge

Consider first the simple case where both the condenser and evaporatorare plug-flow heat exchangers Figure24(b)and25(b)respectively There is no additionalliquid level in the cycle and we neglect the holdup in the piping compressor andvalve We then have that the total charge in the system ismtot = mevaporator+mcondenser An illustration is given in Figure26 The compressor controls the loadof the cycle and the choke valve controls the degree of super-heating We assumethat the flow of hot and cold source are kept constant so there are no degrees offreedom in the cycle However we have indicated an external fillingemptyingsystem that may be used to adjust the charge in the system

TC

Super-heatcontrol

(a) Initial charge gives saturation at condenseroutlet

TC

Super-heatcontrol

(b) Increased charge gives sub-cooling at con-denser outlet

Figure 26 A simple design with external fillingemptying system illustrating thedegree of freedom related to active charge

In Figure26(a)the charge is just enough to give saturation before the choke valveIf the charge is increased by filling from the external tank we will get the situationillustrated in Figure26(b) The explanation is as follows

Since the degree of super-heating out of the evaporator is controlled (by the chokevalve) there will be no room for additional charge in the evaporator (except theincrease due to higher density in vapour phase for higher pressure)The chargewill therefor accumulate in the condenser where it will increase the pressure (andtherefore the heat transfer) and give sub-cooling at the outlet This illustrates thatthere is an extra degree of freedom related to the active charge This degree offreedom is lost if the condenser design in Figure24(a)is used


The active charge may also be controlled without the external tank This is dis-cussed in Chapter3

Remark If we remove refrigerant from the cycle in Figure26(a)we may get to a pointwhere the condenser pressure is to low to get full condensation This illustrates why sepa-rate tank with a variable liquid level is desirable

216 Operation of simple cycles

The simplest (and probably most well known) process using vapour compressioncycle is the household refrigerator The evaporator is mounted inside the refrig-erator and the remaining equipment (compressor condenser and chokevalve) ismounted on the outside of the refrigerator

Household refrigerators are usually controlled with a thermostat that switches thecompressor on and off depending on the temperature inside the refrigerator Theexpansion device is usually fixed for example a capillary tube Natural convec-tion heat exchangers (no fans) are used Thus it is only one manipulated variablenamely the onoff switch for the compressor Design and characteristics ofrefrig-erators is discussed byDossat(2002)

Onoff control of the compressor is reasonable for a refrigerator since the com-pressor will operate at the design point with high efficiency most of the time (notduring start-up) The use of a constant expansion device is also reasonable sincethe conditions are more or less constant (only small variations in the refrigeratorand room temperature) However for other applications it is necessarywith con-tinues capacity control The compressor may then have variable speed andthereare larger variations in the cycle such that a fixed valve position is inefficient Itis therefor normal to also have a variable valve In addition it is common to haveadjustable fans on the heat exchangers to improve heat transfer This gives fourmanipulated variables but as discussed above there may be a fifth manipulatedvariable This fifth manipulated variable is related to the active charge in the cycleand depends on the design of the cycle

The compressor is normally used to control the cooling load The valve may beused to control the liquid level in the evaporator (or the degree of super-heating)As shown in AppendixB it is close to optimal to have the fans at constant speedsWe are then left with one degree of freedom that should optimize the operationGood controlled variables are found in Chapter4 and include the degree of sub-cooling (∆Tsub) and the temperature difference at the condenser outlet A possiblecontrol configuration is shown in Figure27

22 Some design features 13

TC

Super-heatcontrol

Sub-coolingcontrol

Figure 27 A possible control configuration for a simple refrigerant cycle Thefans are at constant speed

22 Some design features

The description in this section is based on refrigeration processes but the sameapplies for heating cycles

221 Constant versus varying temperature loads

The choice of refrigerant and configuration depends heavily on the cooling loadIf the cooling load has a constant or close to constant temperature throughout theevaporator a pure refrigerant will give a good temperature match Therefrigerantwill evaporate at a slightly lower temperature than the cooling load (TC) Thisis illustrated in Figure28(a) where there is some super-heating (∆Tsup) Notethat this super-heating may be removed if the evaporator is designed as a floodedevaporator We have assumed constant temperature loads in Chapter3 and4 bythe use of cross-flow heat exchangers Other cases of constant temperature loadsare single component condensation or evaporation

Often the cooling load change temperature as it is being cooled If the temperaturevaries a lot a single pure refrigerant will give large temperature differences inthe warm end of the evaporator illustrated in Figure28(b) This large temperaturedifference gives a loss in terms of cycle efficiency COP so it may be economicallyattractive to consider alternatives to a single pure refrigerant cycle

14 Introduction to vapour compression cyclesT

[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup

(a) Constant temperature load

T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant

(b) Variable temperature load

Figure 28 Evaporator temperature profile

Using a multicomponent refrigerant will give a gliding temperature also on therefrigerant side in the evaporator This is illustrated in Figure28(b) The refriger-ation composition may be adjusted to optimize the performance of the system

222 Two-stage expansion and two pressure levels

(a) Flowsheet

Large vapour fraction LargeWs

High temperatureHHHHj

P[bar]

h[Jkg-1]

(b) Pressure enthalpy diagram dotted cycle is con-ventional one-stage expansion

Figure 29 Two stage expansion improves cycle efficiency

If a high pressure ratioPR= PhPl

is required (because of large difference inTH andTC) there will be a large vapour fraction in the expansion process All this vapour


has to be compressed back to the high pressure if the simple layout in Figure21isutilized This is illustrated with the dotted cycle in Figure29(b) The large vapouramount will not contribute to the cooling (except in the super-heating section)for a pure refrigerant and only give a minor contribution for mixed refrigerantsAn improvement is to do the expansion in two (or more) stages as illustrated inFigure29(a) The vapour from the intermediate pressure level is fed either to thecompressor as a side stream or between two compressors This configuration hastwo effects making it more desirable

bull The vapour generated in the expansion down to the intermediate pressure isnot expanded further so the necessary compressor power is reduced

bull The vapour from the intermediate pressure level is colder than the vapourthat has been compressed from the lowest pressure level so the mixing ofthetwo streams will work as inter-cooling in the compressor and the necessarycompressor power is reduced as well as the outlet temperature

Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2

(b) Temperature profile

Figure 210 Cooling at two pressure levels

An interesting use of two-stage expansion occur if cooling is needed at severaltemperature levels eg a process stream that is cooled One may then havean ad-ditional evaporator at the intermediate pressure level This configuration isshownin Figure210(a) The process stream is first cooled by the intermediate pressurelevel (Evaporator 1) and then by the low pressure level (Evaporator 2) The pres-sure enthalpy diagram is still as in Figure29(b) but the amount of vapour at theintermediate pressure level is increased The temperature profile in the evaporatorsmay be illustrated by Figure210(b) Control of such cycles are discussed byWil-


son and Jones(1994) and we study degrees of freedom for this process in Chapter8

223 Internal heat exchange

Two ways of implementing internal heat exchange is illustrated in Figure211 Theconfiguration shown in Figure211(a)is common forCO2 cycles (Neksaa et al1998) but may also be used for other working fluid The positive effect is that theexpansion loss is reduced because of the extra sub-cooling before theexpansionvalve However the compressor inlet is heated which is negative for the efficiencyDepending on the working fluid and the operating point this kind of internal heatexchange may improve the efficiency (Radermacher 1989) Also if the suctionline to the compressor will heat the vapour anyhow it is better to use this coolinginternally

The internal heat exchange configuration shown in Figure211(b)is often used inLNG processes Unless the internal heat exchanger may be bypassedit does notcontribute to additional manipulated variables

(a) May improve cycle efficiency also forpure fluids often used forCO2 cycles

(b) No effect for pure refrigerants often usedin LNG processes

Figure 211 Internal heat exchange

224 Cascaded cycles

The configuration in Figure212shows two cycles in cascade both with internalheat exchange The refrigerant in the first cycle is condensed (partly of fully) by an

23 Conclusion 17

HX2

HX1

Figure 212 Cascaded cycles shown with internal heat exchange

external fluid (eg air or water) and then further cooled in HX1 This refrigerantis used as cooling in HX1 after expansion The refrigerant in the secondcycleis cooled and possibly partly condensed with the same external fluid Additionalcooling is provided in both HX1 (by heat exchange with the first refrigerant cycle)and in HX2 The second refrigerant is expanded and vaporized through HX2 toprovide the cooling at the lowest temperature This layout shown in Figure212isoften used for liquefaction of natural gas The number of cycles typicallyvariesfrom one to three and the refrigerant may be mixed or pure (then without internalheat exchange)

23 Conclusion

There are a number of design features that may be utilized in refrigeration pro-cesses based on the vapour compression cycle It is not in the scope ofthis thesisto find which of these features are part of the truly optimal design but rather to startfrom a given process design and study optimal operation However it seems thata study to quantify the energy improvements imposed by different design featuresis needed and may be an issue for further work

18 Bibliography

Bibliography

Barclay M A and Yang C C (2006) Offshore LNG The perfect startingpoint for the 2-phase expanderin lsquoOffshore Technology Conference Hous-ton Texas USArsquo

Dossat R J (2002)Principles of refrigeration Prentice Hall

Langley B C (2002)Heat pump technology Prentice Hall

Neksaa P Rekstad H Zakeri G R and Schiefloe P A (1998) lsquoCO2-heat pumpwater heater characteristics system design and experimental resultsrsquoInt JRefrigeration21 172ndash179

Radermacher R (1989) lsquoThermodynamic and heat-transfer implicationsof work-ing fluid mixtures in Rankine cyclesrsquoInt J Heat Fluid Flow10(2) 90ndash102

Wilson J A and Jones W E (1994) lsquoThe influence of plant designon refriger-ation circuit control and operationrsquoEuropean Symposium on Computer AidedProcess Engineering (ESCAPE) 4 Dublinpp 215ndash221

Chapter 3

Optimal operation of simplerefrigeration cyclesPart I Degrees of freedom andoptimality of sub-cooling

Published in Computers amp Chemical Engineering (2007) 31 pages 712-721

The paper focuses on the operation of simple refrigeration cycles Withequipment given there are from a control and operational point of viewfive steady state degrees of freedom the compressor power the heattransfer in the condenser the heat transfer in the evaporator the chokevalve opening and the active charge in the cycle Different designs foraffecting the active charge including the location of the liquid receiverare discussed With a given load (eg given cooling duty) the com-pressor power is set Furthermore it is usually optimal to maximize theheat transfer The two remaining degrees of freedom (choke valve andactive charge) may be used to set the degree of super-heatingand sub-cooling It is found that super-heating should be minimized whereassome sub-cooling is optimal For a simple ammonia cycle sub-coolinggives savings in compressor power of about 2 In this paperrefriger-ation (cooling) cycles are considered but the same principles apply toheat pumps

19

20 Simple cycles Part I

31 Introduction

Cyclic processes for heating and cooling are widely used and their powerrangesfrom less than 1kW to above 100MW In both cases vapour compression cycle isused to ldquopumprdquo energy from a low to a high temperature level

The first application in 1834 was to produce ice for storage of food which led tothe refrigerator found in most homes (Nagengast 1976) Another well-known sys-tem is the air-conditioner (AC) In colder regions a cycle operating in the oppositedirection the ldquoheat pumprdquo has recently become popular These two applicationshave also merged together to give a system able to operate in both heating andcooling mode

In Figure 31 a schematic drawing of a simple cycle is shown together with atypical pressure-enthalpy diagram for a sub-critical cycle The cycleworks asfollows

The low pressure vapour (4) is compressed by supplying workWs to give a highpressure vapour with high temperature (1) The vapour is cooled to its saturationtemperature in the first part of the condenser condensed in the middle part andpossibly sub-cooled in the last part to give the liquid (2) In the choke valve thepressure is lowered to its original value resulting in a two-phase mixture (3) Thismixture is vaporized and possibly super-heated in the evaporator (4) closing thecycle

QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup

Pmmiddot middot middot middot middot middot middot middot middot

(b)

Figure 31 (a) Simple refrigeration or heat pump cycle with (b) typical pressure-enthalpy diagram indicating both sub-cooling and super-heating

The choke valve may be replaced by an expander for improved efficiency but this

31 Introduction 21

Table 31 Structure of model equationsHeat exchangers (condenser and evaporator)Q = U middot

int∆T dA = mmiddot (houtminushin)

P = Psat(Tsat)m= ρVValvem= zmiddotCV

radic∆Pmiddotρ hout = hin

CompressorWs = m(houtminushin) = mmiddot (hsminushin)η

is not considered here The coefficient of performance for a refrigeration cycle(refrigerator AC) is defined as

COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)

The COP is typically around 3 which indicates that 33 of the heat duty is addedas work (eg electric power)

In this paper the objective is to optimize the operation of a given cycle (Figure31)in terms of maximize the COP or specifically to minimize the compressor powerWs for a given cooling loadQc We consider only steady state operation Themodel equations are summarized in Table31 Note that pressure losses in pipingand equipment are neglected We also assume that the temperature of the hot(TH)and cold (TC) source are constant throughout the heat exchanger This assumptionholds for a cross flow heat exchanger In practice there may be some operationalconstraints for example maximum and minimum pressure constraints which arenot considered here

In industrial processes especially in cryogenic processes such as air separationand liquefaction of natural gas (LNG process) more complex refrigeration cyclesare used in order to improve the thermodynamic efficiencies These modificationslower the temperature differences in the heat exchangers and include cycles withmixed refrigerants several pressure levels and cascaded cycles Our long termobjective is to study the operation of such processes However as a start we needto understand the simple cycle in Figure31

An important result from this study is the degree of freedom analysis given inSection32 We find that the ldquoactiverdquo charge plays an important role in operation ofcyclic processes This is also directly applicable to more complex designs Unlikean open process a closed cyclic process does not have boundary conditions onpressures imposed by the flows in and out of the system Instead the pressure level


is indirectly given by the external temperatures heat exchanger sizesload andthe active chargeThe active charge is defined as the total mass accumulated inthe process equipment in the cycle mainly in the condenser and evaporator butexcluding any adjustable mass in liquid receivers (tanks)

The effect of a change in active charge on operation depends on the specific designIntuitively it seems that an increase in active charge must increase the pressureand indeed this is true in most cases For example this is the case for the mod-els used in this paper with plug-flow in the heat exchangers Then more liquidinthe condenser gives more sub-cooling which effectively reduces cooling and pres-sure increases Similarly more liquid in the evaporator gives less super-heatingeffectively increasing heat transfer and pressure increases However there may bedesigns where the effect of charge on pressure is opposite For example considera well-mixed flooded condenser where the heat transfer coefficientU to liquid islarger than to vapour An increase in charge (liquid) may then improve coolingand pressure decreases In any case the main point is that the ldquoactiverdquo charge is adegree of freedom that affects the operation of the system and this paper focuseson how to use it effectively

Although there is a vast literature on the thermodynamic analysis of refrigerationcycles there are very few authors who discuss their operation and control Somediscussions are found in text books such asStoecker(1998) Langley(2002) andDossat(2002) but these mainly deal with more practical aspectsSvensson(1994)andLarsen et al(2003) discuss operational aspects A more comprehensive recentstudy is that ofKim et al (2004) who consider the operation of trans-criticalCO2

cycles They discuss the effect of ldquoactive chargerdquo and consider alternatives forplacing the receiver

The paper also discuss super-heating and sub-cooling In the literature it is gen-erally taken for granted that there for a given cycle should be no sub-cooling andsuper-heating (∆Tsub= 0C and∆Tsup= 0C) in optimal operation For exampleStoecker(1998 page 57) states that

The refrigerant leaving industrial refrigeration condensers may be slightlysub-cooled but sub-cooling is not normally desired since it indicates thatsome of the heat transfer surface that should be be used for condensation isused for sub-cooling At the outlet of the evaporator it is crucial for protectionof the compressor that there be no liquid so to be safe it is preferable for thevapor to be slightly super-heated

In this study we confirm that super-heating is not optimal The issue of sub-cooling is less clear Of course sub-cooling in itself is always optimal as lessrefrigerant needs to be circulated The issue is whether sub-cooling is optimal

32 Degrees of freedom in simple cycles 23

for a given cold source temperature and a given condenser area because sub-cooling will reduce the temperature driving forces which must be compensatedby increasing the pressure We find contrary to popular belief that withgivenequipment sub-cooling in the condenser may give savings in energy usage (com-pressor power) in the order of 2 An ammonia case study is presented to obtainnumerical results

32 Degrees of freedom in simple cycles

321 Design versus operation

Table 32 shows typical specifications for the simple refrigeration cycle in Fig-ure 31 in design (find equipment) and in operation (given equipment) The fivedesign specifications include the load the two pressures and the degreeof sub-cooling and super-heating Based on these five design specifications external con-ditions and an assumed isentropic efficiency for the compression we may obtainthe following fourequipment parameters which can be adjusted during operationcompression work (Ws) valve opening (z) and effective heat transfer (includingUA-values) for the two heat exchangers Initially we were puzzled because wecould not identify the missing fifth equipment parameter to be adjusted during op-eration However we finally realized that we can manipulate the rdquoactive chargerdquoin the cycle which affects the operation The fact that the charge is an indepen-dent variable is unique for closed systems since there is no (external) boundarycondition for pressure which would otherwise set the active charge

Table 32 Typical specifications in design and operationGiven

Design Load (egQh) Pl Ph ∆Tsup and∆Tsub 5Operation Ws (load) choke valve opening (z)

effective heat transfer (egUA) in twoheat exchangers and active charge 5

322 Active charge and holdup tanks

For the simple cycle in Figure31we have the following overall material balance

mtot = mevap+mcon+mvalve+mcomp︸︷︷︸

mactive

+mtanks (32)


Normally the holdups in the valve and compressor are neglected and we get

mtot = mevap+mcon︸︷︷︸

mactive

+mtanks (33)

With no filling emptying or leaks the total massmtot is fixed We have not in-cluded a holdup tank in Figure31 but in practice it is common to include a tankor receiver with variable liquid mass It is assumed that a change inmtanks (egby filling or leaking) with a constant active charge (mactive) does not affect the op-eration of the cycle This implies that the tank must contain both liquid and gasin equilibrium (saturated) Then we can move mass to or from the tank withoutaffecting the pressure and thus without affecting the rest of the cycleThus theliquid tank makes operation independent of the total charge in the system

More importantly the extra tank introduces an additional degree of freedom Thiscan be seen from Equation33 With mtot constant we can by changing the mass(liquid) in the tank (mtank) change the active charge (mactive) This shows thatmtank

has an indirect steady state effect on the active charge and can therefore be usedfor control purposes of course provided that we have means of changing it

Although it is possible to introduce several tanks in a cycle we only have onematerial balance for each cycle so from Equation33this will not add any steady-state degrees of freedom with respect to the active charge

Rule 31 In each closed cycle we have one degree of freedom related to the activecharge which may be indirectly adjusted by introducing a variable liquid level(tank receiver) in the cycle

Rule 32 In each closed cycle there will be one liquid holdup that does not needto be explicitly controlled because the total mass is fixed This is usually selectedas the largest liquid volume in the closed system The remaining liquid levels(holdups) must be controlled (to avoid overfilling or emptying of tanks)

Remark 1 Note that in Rule32it says ldquodoes not needrdquo rather than ldquomust notrdquo Thus Rule32does not say that we cannot control all the liquid volumes in the system (including thelargest one) but it just states that it is not strictly necessary In fact controlling all theliquid volumes provides a way for explicitly controlling the active charge in the cycle(Rule31)

Remark 2 Introducing additional liquid tanks may be useful for operation but at leastfor pure fluids these will not introduce any additional steady-state degrees of freedombecause we can move mass from one tank to another without affecting operation Alsoto avoid that tanks fill up or empty these additional levels must be controlled (Rule32)either by self-regulation or feedback control


Remark 3 In mixed refrigerantcycles two tanks may be used to indirectly change thecomposition of the circulating refrigerant In this case the two tanks have different compo-sition so moving mass from one tank to another does affect operation For more complexcycles the maximum number of degrees of freedom related to tank holdups is the numberof components in the refrigerant

Adjusting the active charge

In order to freely adjust the active charge we need to introduce a liquid tank (re-ceiver) plus an extra valveKim et al (2004) discuss alternative locations for thevariable tank holdup (liquid receiver) In Figure32 we show cycles for the twomain cases where the tank is placed (a) on the high pressure side after the con-denser and (b) on the low pressure side after the evaporator Other placements andcombinations are possible but these are only variations of these two and willnotadd any steady-state degrees of freedom for pure refrigerants

The most obvious way of introducing a means for adjusting the tank holdup is toadd an extra valve before the tank as shown in Figure32 In Figure32(a) the

z

QC

QH

Ws

Pl

Ph

Pm

(a) Liquid tank and extra valve on high pres-sure side

z

QC

QH

Ws

Pl

Ph

(b) Liquid tank and extra (non-optimal)valve on low pressure side

Figure 32 Simple cycle with variable active charge

liquid tank is located at an intermediate pressurePm after the condenser Sincethe extra valve is on the ldquosame siderdquo as the expansion valve (choke) the pressuredrop over the extra valve will not effect the efficiency of the cycle Since Pm isassumed to be the saturation pressure at the tank temperature the exit stream fromthe condenser must be sub-cooled (or super-critical but this is not considered inthis paper) Thus in Figure32(a) the pressure drop across the valve may be used


to adjust the degree of sub-cooling in the condenser To understand how the extravalve creates sub-cooling consider the pressure-enthalpy diagram inFigure31The receiver (tank) with saturated liquid operates at saturation pressure Pm andthe pressure drop for the extra valve introduces a pressure dropPhminusPm As seenfrom Figure31 the corresponding operating point 2 at the exit of the condensermust then be at a sub-cooled state

Another possibility is to place the tank after the evaporator as shown in Figure32(b) With this design the stream exiting the evaporator is not fully evapo-rated and by lowering the pressure through the extra valve the vapour exiting thevalve becomes saturated (see pressure-enthalpy diagram) Howeverin this casethe valve introduces a pressure drop that must be compensated by increasing thecompression power so a valve here is generally not optimal

A low pressure tank may not be desirable from a practical point of view since thevapour velocity will be highest at this point in the cycle and the extra equipmentwill increase the pressure drop

Extra valve removed

An extra valve is generally required to freely adjust the active charge However inmany practical cases the extra valve in Figure32(a)and32(b)is removed Whateffect does this have

bull High pressure tank without valve Without the valve we have at steady statethe same thermodynamic state at the exit of the condenser as at the exitfrom the tank Thus the exiting stream from the condenser will be satu-rated liquid The most common design is shown in Figure33 where thetank and condenser are merged together so that the saturated liquid from thecondenser drains into the receiver As we will show this is not generallyoptimal Thus in this design we have used a degree of freedom (ldquofully openvalverdquo) to set the degree of sub-cooling to zero (not optimal)

bull Low pressure tank without valve (Figure34(a)) This gives saturated vapourto the compressor Fortunately this is generally optimal for the cycle as awhole because the inlet temperature to the compressor should be as low aspossible to minimize vapour volume and save compression power Thus inthis design we have used a degree of freedom (ldquofully open valverdquo) to set thedegree of super-heating to zero (optimal) Two designs are shown in Figure34(a) one with a separate receiver and one using a flooded evaporator Thedesigns are equivalent thermodynamically but the heat transfer coefficient


QH

Figure 33 Condenser with saturation at outlet giving no sub-cooling (commondesign but non-optimal)

and pressure drop will be different

In summary removing the valve gives saturation at the exit of the heat exchangerIn the case of high-pressure liquid tank we get a sub-optimal design if we removethe valve whereas for the low-pressure tank we get an optimal design if the extravalve is removed

QCTC

(a) With separate receiver

QC

(b) Flooded evaporator

Figure 34 Evaporator with saturation at outlet giving no super-heating (optimal)

323 Degrees of freedom for operation

In summary we have the following five operational or control degrees offreedomfor a simple refrigeration cycle (Figure31)


1 Compressor powerWs We assume here that it is used to set the ldquoloadrdquo forthe cycle

2 3 Effective heat transfer There are two degrees of freedom related to adjust-ing the heat transferred in the condenser and evaporator This may be donein many ways for example by introducing bypasses changing the flowratesof coolant or using a flooded condenser or evaporator to change the effec-tive UA-value However we generally find that it is optimal to maximizethe effective heat transfer in the condenser and evaporator Thereare ex-ceptions where it may not be optimal to maximize the heat transfer in thecondenser and evaporator for example because of costs related to pumpsfans or coolants but these degrees of freedom are not consideredin the fol-lowing

4 Choke valve opening (z)

5 Active charge (see Section322)

In practice we are then with a given load and maximum heat transfer left withtwosteady state degrees of freedom These are the choke valve opening (z) and theactive charge (mactive) These may be used to set the degree of super-heating anddegree of sub-cooling The pressure levels (Ph andPl ) are indirectly determined bythe given (maximum) value of the heat transfer

33 Discussion of some designs

As discussed in more detail in Section34 we find that the thermodynamic effi-ciency is optimized by having no super-heating and some sub-cooling With thisin mind we next discuss some alternative designs

331 Optimal designs

Two potentially optimal designs are shown in Figure35 The reason we say ldquopo-tentially optimalrdquo is because they will only be optimal if we use the optimal valuefor the sub-cooling and super-heating

To avoid super-heating we have in Figure35(a)and35(b)a low-pressure tank(receiver) after the evaporator This tank will give saturated vapourout of theevaporator at steady state (optimal) and also by trapping the liquid it will avoidthat we get liquid to the compressor during transient operation To avoid super-heating we must have vapour-liquid equilibrium in the tank This may be achieved

33 Discussion of some designs 29

replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph

(a) Optimal with 1 tank

QC

QH

WsSub-coolingcontrol

z

LC

Pl

Ph

(b) Optimal with 2 tanks

Figure 35 Two potentially optimal designs with sub-cooling and no super-heating

by letting the vapour bubble through the tank An alternative design is the floodedevaporator in Figure34(b)

At the high-pressure side we show optimal designs with both (a) no receiver and(b) a receiver and an extra valve In (a) the choke is used to control the degree ofsub-cooling (∆Tsub) Also other control policies are possible for example keep-ing the choke valve position at its optimal value or controlling the pressure butcontrolling ∆Tsub was found byJensen and Skogestad(2005) to be a good self-optimizing controlled variable

The design in Figure35(b)is thermodynamically equivalent to Figure35(a) butthe addition of the tank may prevent that we get two-phase flow with vapour ldquoblowoutrdquo through the choke We here have two adjustable holdups so from Rule 32one of them must be controlled In Figure35(b)is shown the case where the chokevalve is used to control the level in the high pressure tank but alternatively it couldcontrol the level in the low pressure tank

332 Non-optimal designs

Three non-optimal designs are shown in Figure36 Figure36(a)shows the designused in most applications except that the tank and condenser are often integrated asshown in Figure33 This common design has two errors compared to the optimalsolution 1) There is no sub-cooling in the condenser and 2) there is super-heatingin the evaporator The super-heat control is in practice accomplished witha ther-mostatic expansion valve (TEV) In theory one could get optimality by setting the


setpoint for super-heating to zero but in practice this is not possible because thiscould give liquid out of the evaporator The setpoint for super-heatingis typicallyabout 10C

In Figure36(b)we have two liquid tanks one after the evaporator and one after thecondenser This design is better since there is no super-heating in the evaporatorbut one error remains There is no sub-cooling in the condenser Note that we needto control one of the liquid levels in accordance with Rule32

Another non-optimal design is shown in Figure36(c) Here we have introducedthe possibility for sub-cooling but we have super-heating which is generally not

QC

QH

Ws

Super-heatcontrolz

Pl

Ph

(a) Non-optimal 1 This design has two er-rors 1) No sub-cooling and 2) Super-heating

QC

QH

Ws

z

LC

Pl

Ph

(b) Non-optimal 2 This design has one errorNo sub-cooling

QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph

(c) Non-optimal 3 This design has one errorSuper-heating

Figure 36 Three non-optimal designs

34 Optimality of sub-cooling 31

optimal

34 Optimality of sub-cooling

We have several times made the claim that sub-cooling may be optimal To justifythis somewhat controversial claim we start by considering a specific example

341 Ammonia case study

The objective is to cool a storage building by removing heat (QC) as illustrated inFigure37 The cycle operates between a cold medium of air inside the building(TC = Troom) and hot medium of ambient air (TH = Tamb) The steady state heatloss from the building is 20kW and the cooling loadQC is indirectly adjusted bythe temperature controller which adjusts the compressor work (Ws) to maintainTC = Ts

C

TC TsC

QH

QC

z

Ws

Ph

Pl

Figure 37 Cold warehouse with ammonia refrigeration unit

Some data for the cycle

bull Ambient temperatureTH = 25C

bull Indoor temperature setpointTsC = minus12C

bull Isentropic efficiency for compressor is 95


bull Heat transfer coefficients (U) are 1000 and 500 Wm-2K-1 for the evaporatorand condenser respectively

bull Heat exchangers with areas given in Table33

bull Thermodynamic calculations based on SRK equation of state

The equipment is given and we have 5 steady-state operational degreesof free-dom (Section32) With a given load and maximum heat transfer we have tworemaining steady state degrees of freedom which may be viewed as the degree ofsub-cooling (∆Tsub) and the degree of super-heating (∆Tsup) The performance ofthe cycle measured by the compressor powerWs was optimized with respect tothe two degrees of freedom We find as expected that super-heating is not optimalbut contrary to popular belief the results in Table33 show that sub-cooling by466C reduces the compression workWs by 174 compared to the case withsaturation out of the condenser The high pressurePh increases by 045 but thisis more than compensated by a 212 reduction in flowrate The sub-cooling in-creases the condenser chargeMcon by 501 Figure38shows the correspondingpressure enthalpy diagram for the two cases and Figure39shows the temperatureprofile in the condenser Similar results are obtained if we use other thermody-namic data if we change the compressor efficiency or if we let UA be smaller inthe sub-cooling zone

Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

(a) Optimal operation without sub-cooling (Fig-ure36(b))

Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

(b) Optimal operation with sub-cooling allowed(Figure35)

Figure 38 Pressure-enthalpy diagrams with and without sub-cooling

The improvement of 2 would be larger if the pressure drop in the piping andequipment was accounted for in the model because the mass flowrate is reducedwith an unchanged low pressure Thus the volumetric flowrate in the low pressure


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120

(a) Temperature profile without sub-cooling

Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120

(b) Temperature profile with sub-cooling

Figure 39 Temperature profile in condenser

Table 33 Optimal operation with and without sub-coolingNo sub-cooling Optimal sub-cooling

Ws[W] 4648 4567QC [kW] 20 20m[kgs-1] 00177 00173Mcon

lowast [kg] 0301 0316∆Tsub[

C] 000 466∆Tsup[

C] 000 000∆Tmin con[

C] 500 0491Ph [bar] 1163 1168Pl [bar] 217 217Acon[m2] 870 870Avap[m2] 400 400

lowastEvaporator charge has no effect because of saturation (no super-heating) in the evaporator

side is reduced and this is important as pressure drop is most critical at low pres-sure The pressure losses on the high pressure side will also be slightly reduced(because of smaller flowrate and higher pressure) but this is less important for theefficiency of the cycle


342 Explanation

The irreversible isenthalpic expansion through the choke valve gives a thermody-namic loss The reason for the improvement in efficiency by sub-cooling is thatloss is reduced because less vapour is formed see Figure38 This more than com-pensates the increased irreversible loss due to larger temperature difference in thecondenser To understand this in more detail consider Figure310which shows aconceptual pressure enthalpy diagram of a typical vapour compression cycle Wehave indicated a cycle without sub-cooling (solid line) and the same cycle withsub-cooling (dotted line) Note that since we in the latter case have a higher con-denser pressure (and therefore also a higher temperature in the condensing section)we will with given equipment (UA-values) have more heat transfer whichgivesa lower outlet temperature The condenser outlet will follow the line ldquoCon outrdquowith increasing pressure The line will asymptotically approach the hot sourcetemperatureTH and we want to find the optimal operating point on this line

P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out

Figure 310 Pressure-enthalpy diagram for a cycle with and without sub-cooling

If we consider moving from one operating point to another we require an increasein the COP for the change to be optimal

∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)

COPmiddot∆ws lt ∆qC (35)


whereqC middot m= QC andws middot m= Ws We assume thatQC [Js-1] is given and thatm[kgs-1] andqC [Jkg-1] may vary We use∆Tsub as the independent variable andintroduce differentials The requirement for improving efficiency is then fromEquation35

(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)

According to Equation36 for an initial COP of 3 the increase in specific dutyin the evaporator (∆qC) should be 3 times larger than the increase in specific com-pressor power (∆ws) to give improved performance In Figure310we have that∆qC asymp ∆ws so the optimal degree of sub-cooling is clearly less than that indi-cated by this figure Note however that the ldquoCon outrdquo line is much flatter forsmaller∆qC so a small degree of sub-cooling may be optimal The optimum islocated at the degree of sub-cooling where the inequality in Equation36becomesan equality In the case study we found that the optimum outlet temperature fromthe condenser (2549C) is closer toTH (25C) than the saturation temperature(3015C)

Similar considerations on optimizing the pressurePh have been made earlier fortrans-criticalCO2-cycles (Kim et al 2004) However for sub-critical cycles likethe ammonia cycle studied above it has been assumed that the pressure is fixed bya saturation condition

343 Discussion of sub-cooling Why not found before

The above results on optimality of sub-cooling is contrary to previous claims andpopular belief Why has this result not been found before

Reason 1 Not allowed by design

The design of the condenser is often as shown in Figure33 where the saturatedliquid drains into a liquid receiver In this design it is not possible to have sub-cooling

Reason 2 Infinite area case

The optimal degree of sub-cooling becomes smaller as we increase the heattrans-fer (UA-values) In particular with an infinite heat transfer area sub-cooling is


not optimal In this case the temperature at the condenser outlet is equal to thehotsource temperatureTH Neglecting the effect of pressure on liquid enthalpy theenthalpy is also given We then find that∆qC = 0 and sub-cooling is not optimalas illustrated in Figure311

P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line

Figure 311 Pressure-enthalpy diagram for infinite area case where condenseroutlet is at hot source temperatureTH

In practice the enthalpy depends slightly on pressure (as indicated by thecurvedconstant temperature lines in Figure311) so ∆qC might be larger than zero butthis effect is too small to change the conclusion that sub-cooling is non-optimalwith infinite area

Reason 3 Specifying HRAT

The minimum approach temperature (∆Tmin or HRAT) is commonly used as a spec-ification for design of processes with heat exchangers The idea is to specify ∆Tmin

in order to get a reasonable balance between minimizing operating (energy)costs(favored by a small∆Tmin) and minimizing capital costs (favored by a large∆Tmin)Although specifying∆Tmin may be reasonable for obtaining initial estimates forstream data and areas it should not be used for obtaining optimal design data -and especially not stream data (temperatures) This follows because specifying∆Tmin results in an optimum with no sub-cooling This can be seen by letting theTH-line in Figure311representTH + ∆Tmin The condenser outlet temperature isthenTH +∆Tmin and similarly to the infinite area case we get∆qC = 0 (neglectingthe effect of pressure on liquid enthalpy) and sub-cooling is not optimal

The results can also be understood because specifying∆Tmin favors designs with∆T being as close as possible to∆Tmin throughout the heat exchanger and this

35 Discussion 37

clearly disfavour sub-cooling see Figure39(b)

A third way of understanding the difference is that we end up with two differentoptimization problems for design (Equation37) and operation (Equation38)

min (Ws) (37)

subject to TCminusTsC = 0

∆Ti minus∆Tmini ge 0

min (Ws) (38)


Amaxi minusAi ge 0

For the ammonia case study solving37with ∆Tmin = 5C gives the data for ldquoNosub-coolingrdquo in Table33 Setting the resulting areas asAmax and solving theoptimization problem38results in A=Amaxand the data for ldquoOptimal sub-coolingrdquoin Table33 We see that specifying∆Tmin gives no sub-cooling whereas fixingthe heat exchanger areas to the same value gives 466C of sub-cooling

35 Discussion

351 Super-heating by internal heat exchange

For the simple cycle in Figure31 some sub-cooling in the condenser was foundto be optimal and we here discuss whether other means of obtaining further sub-cooling in particular the use of internal heat exchange (Figure312) may be ben-eficial

Consider first the case when the vapour leaving the evaporator is saturated In thiscase the internal heat exchange in Figure312has no effect on the overall processat least for pure fluids This can be understood because there is no effect on thepressure-enthalpy diagram

Next consider the case where the vapour is super-heated which was previouslywithout internal heat exchange found to be non-optimal Depending on the prop-erties of the fluid this design may be desirable in some cases even for purere-frigerants (Radermacher 1989) In the ammonia case study presented above it isnot optimal with internal heat exchange but for a trans-criticalCO2 cycle internalheat exchange with super-heating is optimal (Neksaa et al 1998)


z

QC

QH

QA

Ws

Pl

Ph


352 Selection of controlled variable

Without internal heat exchange we have found that it is generally optimalto haveno super-heating (∆Tsup= 0C) and some sub-cooling (∆Tsubgt 0C) In practiceno super-heating is easily obtained by use of a design with a low pressure tankas shown in Figure34(a)and Figure35 It is less clear how to get the rightsub-cooling In Figure35 we show a strategy where a valve is used to controlthe degree of sub-cooling∆Tsub However the optimal value of∆Tsub will varyduring operation and also∆Tsub may be difficult to measure and control so it isnot clear that this strategy is good More generally we could envisage anon-lineoptimization scheme where one continuously optimizes the operation (maximizesCOP) by adjusting the valves However such schemes are quite complex andsensitive to uncertainty so in practice one uses simpler schemes like the oneinFigure35 where the valve controls some other variable Such variables could be

bull Choke valve position setpointzs (that is the valve is left in a constant posi-tion)

bull High pressure (Ph)

bull Low pressure (Pl )

bull Temperature out of condenser (T2)

bull Degree of sub-cooling (∆Tsub= T2minusTsat(Ph))

bull Temperature out of evaporator (T4)

bull Degree of super-heating (∆Tsup= T4minusTsat(Pl ))

bull Liquid level in storage tank (to adjust charge to rest of system)

36 Conclusion 39

bull Pressure drop across the extra valve if the design in Figure35(b)is used

The objective is to achieve ldquoself-optimizingrdquo control where a constant setpoint forthe selected variable indirectly leads to near-optimal operation (Skogestad 2000)The selection of ldquoself-optimizingrdquo controlled variables for simple refrigeration cy-cles is the main topic in Part II (Jensen and Skogestad 2007)

36 Conclusion

The ldquoactive chargerdquo in a closed cycle has a steady state effect This is unlike opensystems where we have boundary conditions on pressure To adjust the degreeof freedom related to the ldquoactive chargerdquo one needs a liquid tank (receiver) in thecycle The key to make efficient use of this degree of freedom is to allow forsub-cooling in the condenser Conventional wisdom says that one should avoidsub-cooling in the condenser to maximize the efficiency However we find thatsome sub-cooling is desirable For the ammonia case study we get savings in theorder of 2 by using the design in Figure35 that allows for sub-cooling Thesavings would be even larger if we compared with the common design in Figure36(a)which in addition to having no sub-cooling also gives super-heating

Nevertheless the savings in themselves are not very large More importantly theresults show that the active charge is a degree of freedom and that the sub-coolinggives some decoupling between the high pressurePh and the hot source temper-atureTH This is similar to that found for other cycles including mixed (multicomponent) fluids and trans-criticalCO2

Frictional pressure drops in the equipment have been neglected but their inclusionwould further favor sub-cooling which has a smaller mass flow

Bibliography


Jensen J B and Skogestad S (2005) Control and optimal operation of simpleheat pump cyclesin lsquoEuropean Symposium on Computer Aided Process Engi-neering (ESCAPE) 15 Barcelonarsquo

Jensen J B and Skogestad S (2007) lsquoOptimal operation of a simple refrigerationcycles Part II Selection of controlled variablesrsquoComput Chem Eng31 1590ndash1601

40 Bibliography

Kim M Pettersen J and Bullard C (2004) lsquoFundamental process and systemdesign issues in CO2 vapor compression systemsrsquoProgress in energy and com-bustion science30 119ndash174


Larsen L Thybo C Stoustrup J and Rasmussen H (2003) Control meth-ods utilizing energy optimizing schemes in refrigeration systemsin lsquoEuropeanControl Conference (ECC) Cambridge UKrsquo

Nagengast B (1976) lsquoThe revolution in small vapor compression refrigerationrsquoAmerican Society of Heating Refrigerating and Air-Conditioning Engineers(ASHRAE)18(7) 36ndash40

Neeraas B O Brendeng E Wallentinsen A and Mangersnes M (2001) A newconcept for small-scale LNG productionin lsquoAIChE Spring National Meetingrsquo



Skogestad S (2000) lsquoPlantwide control the search for the self-optimizing controlstructurersquoJ Process Contr10(5) 487ndash507

Stoecker W F (1998)Industrial refrigeration handbook McGraw-Hill

Svensson M C (1994) Studies on on-line optimizing control with applicationto a heat pump PhD thesis Norwegian University of Science and TechnologyTrondheim

Chapter 4

Optimal operation of simplerefrigeration cyclesPart II Selection of controlledvariables


The paper focuses on operation of simple refrigeration cycles and con-siders the selection of controlled variables for two different cycles Oneis a conventional sub-critical ammonia refrigeration cycle and the otheris a trans-criticalCO2 refrigeration cycle There is no fundamental dif-ference between the two cycles in terms of degrees of freedomand op-eration However in practical operation there are differences For theammonia cycle there are several simple control structuresthat give self-optimizing control that is which achieve in practice close-to-optimaloperation with a constant setpoint policy For theCO2 cycle on theother hand a combination of measurements is necessary to achieve self-optimizing control

41 Introduction

Refrigeration and heat pump cycles are used both in homes cars and in industryThe load and complexity varies from small simple cycles like a refrigerator or air-conditioner to large complex industrial cycles like the ones used in liquefactionof natural gas

41

42 Simple cycles Part II

TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl

Figure 41 Simple refrigeration cycle studied in this paper (shown for the ammo-nia case)

The simple refrigeration process illustrated in Figure41is studied in this paper InPart I (Jensen and Skogestad 2007b) we showed that the cycle has five steady-statedegrees of freedom the compressor power the heat transfer in the condenser theheat transfer in the evaporator the choke valve opening and the ldquoactive chargerdquoDifferent designs for affecting the active charge including the locationof the liquidstorage were discussed in Part I

It was found in Part I that there are normally three optimally active constraintsmaximum heat transfer in condenser maximum heat transfer in evaporatorandminimum (zero) super-heating The cycle in Figure41obtains the latter by havinga liquid receiver before the compressor which gives saturated vapourentering thecompressor In addition we assume that the load (eg cooling duty) is specifiedThere is then one remaining unconstrained steady-state degree of freedom relatedto the outlet temperature of the condenser which should be used to optimize theoperation The main theme of the paper is to select a ldquoself-optimizingrdquo controlledvariable for this degree of freedom such that a constant setpoint policy(indirectly)achieves near-optimal operation

We consider two systems

bull a conventional sub-critical ammonia cycle for cold storage (TC = minus10C)

bull a trans-criticalCO2 cycle for cooling a home (TC = 20C)

TheCO2 cycle is included since it always has an unconstrained degree of freedomthat must be used for control This is because there is no saturation condition on the

42 Selection of controlled variable 43

high pressure side which is usually said to introduce one extra degree offreedomto the cycle (Kim et al 2004) However as shown in Part I (Jensen and Skogestad2007b) this ldquoextrardquo degree of freedom is also available in a conventional sub-critical cycle if we allow for sub-cooling in the condenser The sub-cooling willto some extent decouple the outlet temperature and the saturation pressure inthecondenser More importantly some sub-cooling is actually positive in terms ofthermodynamic efficiency (Jensen and Skogestad 2007b) The ammonia cycle isincluded to show that there are no fundamental differences between a sub-criticaland a trans-critical cycle There is some confusion in the literature on this

Although there is a vast literature on the thermodynamic analysis of closed refrig-eration cycles there are few authors who discuss the operation and control of suchcycles Some discussions are found in text books such asStoecker(1998) Lang-ley (2002) andDossat(2002) but these mainly deal with more practical aspectsSvensson(1994) and Larsen et al(2003) discuss operational aspects A morecomprehensive recent study is that ofKim et al(2004) who consider the operationof trans-criticalCO2 cycles

This paper considers steady-state operation and the objective is to find which con-trolled variables to fix The compressor power is used as the objective function(costJ = Ws) for evaluating optimal operation


We consider here the simple cycle in Figure41 where the liquid receiver on thelow pressure side ensures that the vapour entering the compressor is saturatedNote that there is no liquid receiver after the condenser and thus no assumptionof having saturated liquid at the condenser outlet Furthermore it is assumed thatthe heat transfer in both the condenser and evaporator are maximized Finally atemperature controller on the stream to be cooled (here the building temperatureTC) is used to adjust the compressor power

There then remains one unconstrained degree of freedom (choke valve positionz)which should be used to optimize the operation for all disturbances and operatingpoints We could envisage an real-time dynamic optimization scheme where onecontinuously optimizes the operation (minimize compressor power) by adjustingzHowever such schemes may be quite complex and sensitive to uncertainty Theseproblems can be reduced by selecting a good control variable and ideallyone geta simple constant setpoint scheme with no need for real-time optimization Whatshould be controlled (and fixed at least on the short time scale) Some candidates


are

bull Valve positionz(ie an open-loop policy where the valve is left in a constantposition)



bull Temperature out of compressor (T1)

bull Temperature before valve (T2)

bull Degree of sub-cooling in the condenserlowast (∆Tsub= T2minusTsat(Ph))

bull Temperature approach in hot source heat exchanger (T2minusTH)


bull Degree of super-heating in the evaporatordagger (∆Tsup= T4minusTsat(Pl ))

bull Liquid level in the receiver (Vl ) to adjust the active charge in the rest of thesystem

bull Liquid level in the condenser (Vl con) or in the evaporator (Vl vap)

bull Pressure drop across the ldquoextrardquo valve in Figure411dagger

The objective is to achieve ldquoself-optimizingrdquo control where a constant setpoint forthe selected variable indirectly leads to near-optimal operation (Skogestad 2000)Note that the selection of a good controlled variable is equally important in an ldquoad-vancedrdquo control scheme like MPC which also is based on keeping the controlledvariables close to given setpoints

The selection of controlled variables is a challenging task especially if one con-siders in detail all possible measurements so we will first use a simple screeningprocess based on a linear model

421 Linear analysis

To find promising controlled variables the ldquomaximum gainrdquo rule (Halvorsen et al2003) will be used For the scalar case considered in this paper the rule isPrefer controlled variables with a large scaled gain|Gprime| from the input (degree of

lowastNot relevant in theCO2 cycle because of super-critical high pressuredaggerNot relevant for our design (Figure41)


freedom) to the output (controlled variable)Procedure scalar case

1 Make a small perturbation in each disturbancesdi and re-optimize the oper-ation to find the optimal disturbance sensitivitypart∆yoptpartdi Let ∆di denotethe expected magnitude of each disturbance and compute from this the over-all optimal variation (here we choose the 2-norm)

∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2

2 Identify the expected implementation errorn for each candidate controlledvariabley (measurement)

3 Make a perturbation in the independent variablesu (in our caseu is the chokevalve positionz) to find the (unscaled) gainG = ∆y∆u

4 Scale the gain with the optimal span (spanyequiv ∆yopt+n) to obtain for eachcandidate output variabley the scaled gain|Gprime| = |G|

spany

The worst-case lossL = J(ud)minusJopt(ud) (the difference between the cost with aconstant setpoint and re-optimized operation) is then for the scalar case (Skogestadand Postlethwaite 2005 page 394)

L =|Juu|

21

|Gprime|2(41)

whereJuu = part 2Jpartu2 is the Hessian of the cost functionJ In our caseJ = Ws

(compressor work) Note thatJuu is the same for all candidate controlled variablesy

The most promising controlled variables should then be tested on the non-linearmodel using realistic disturbances to check for non-linear effects including feasi-bility problems

422 Combination of measurements

If the losses with a fixed single measurement are large as for theCO2 case studythen one may consider combinations of measurements as controlled variablesThesimple null space method (Alstad and Skogestad 2007) gives a linear combinationwith zero local loss for the considered disturbances

c = h1 middoty1 +h2 middoty2 + (42)


The minimum number of measurementsy to be included in the combination isny = nu+nd In our casenu = 1 and if we want to consider combinations ofny = 2measurements then onlynd = 1 disturbance can be accounted exactly for With theldquoexact local methodrdquo (Halvorsen et al 2003) or the ldquoextended null space methodrdquo(Alstad and Skogestad 2007) it is possible to consider additional disturbancesThe local loss is then not zero and we will minimize the 2-norm of the effect ofdisturbances on the loss


The cycle operates between air inside a building (TC = Troom = minus10C) and am-bient air (TH = Tamb = 20C) This could be used in a cold storage building asillustrated in Figure41 The heat loss from the building is

Qloss= UAloss(TH minusTC) (43)

The nominal heat loss is 15kW The temperature controller shown in Figure41maintainsTC = minus10C and will indirectly giveQC = Qloss at steady-state

431 Modelling

The structure of the model equations are given in Table41and the data are givenin Table42 The heat exchangers are modelled assuming ldquocross flowrdquo with con-stant temperature on the air side (TH = 20C andTC = minus10C) The isentropicefficiency for the compressor is assumed constant The SRK equation of state isused for the thermodynamic calculations The gPROMS model is available on theinternet (Jensen and Skogestad 2007a)






43 Ammonia case study 47

Table 42 Data for the ammonia case studyTH = 20CTC = Ts

C = minus10CCondenser (UA)C = 2500WK-1

Evaporator (UA)E = 3000WK-1

Compressor isentropic efficiencyη = 095Choke valveCV = 00017m2

Building UAloss= 500WK-1

432 Optimal steady-state operation

At nominal conditions the compressor power was minimized with respect to thedegree of freedom (z) The optimal results are given in Table43 and the corre-sponding pressure enthalpy diagram and temperature profile in the condenser areshown in Figure42 Note that the optimal sub-cooling out of the condenser is58C This saves about 20 in compressor power (Ws) compared to the conven-tional design with saturation

h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1

(a) Pressure enthalpy diagram

Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)

(b) Temperature profile in condenser

Figure 42 Optimal operation for the ammonia case study

433 Selection of controlled variables

There is one unconstrained degree of freedom (choke valve openingz) whichshould be adjusted to give optimal sub-cooling in the condenser We want tofind a good controlled variable (see Section42 for candidates) to fix such that


Table 43 Optimal steady-state for ammonia case studyWs[kW] 2975

z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127

∆Tsub= T2minusTsat(Ph) [C] 580T1 [C] 1026T2 [C] 209T3 [C] -150T4 [C] -150

we achieve close-to-optimal operation in spite of disturbances and implementationerror (ldquoself-optimizing controlrdquo)

Linear analysis of alternative controlled variables

The following disturbance perturbations are used to calculate the optimal variationin the measurementsylowast

d1 ∆TH = plusmn10C

d2 ∆TsC = plusmn5C

d3 ∆UAloss= plusmn100WK-1

The assumed implementation error (n) for each variable is given in Table44whichalso summarizes the linear analysis and gives the resulting scaled gains in orderfrom low gain (poor) to high gain (promising)

Some notes about Table44

bull Pl andT4 have zero gains and cannot be controlled The reason for the zerogains are that they both are indirectly determined byQloss

Qloss= QC = (UA)C (T4minusTC) and Pl = Psat(T4) (44)

lowastIn order to remain in the linear region the optimal variations were computedfor a disturbanceof magnitude 1100 of this and the resulting optimal variations were then multiplied by 100 to get∆yopt(di)


Table 44 Linear ldquomaximum gainrdquo analysis of candidate controlled variablesy forammonia case study

|∆yopt(di )|Variable (y) Nom G d1 (TH ) d2 (TC) d3 (UAloss) |∆yopt| n span y |Gprime|Pl [bar] 235 000 0169 0591 0101 0623 0300 0923 000T4 [C] -150 000 0017 0058 0010 0061 100 106 000∆Tsup[

C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[

C] 580 -34078 213 108 108 262 150 412 828Vl con[m3] 067 -57 58e-03 24e-03 14e-03 00064 005 0056 1010T2minusTH [C] 089 -28795 0375 0174 0333 0531 150 203 1418

bull The degree of super-heating∆Tsup can obviously not be controlled in ourcase because it is fixed at 0C (by design of the cycle)

bull The loss is proportional to the inverse of squared scaled gain (see Equation41) This implies for example that a constant condenser pressure (Ph)which has a scaled gain of 337 would result in a loss in compressor powerJ = Ws that is(828337)2 = 603 times larger than a constant sub-cooling(∆Tsub) which has a scaled gain of 828

bull The simple policies with a constant pressure (Ph) or constant valve position(z) are not promising with scaled gains of 337 and 703 respectively

bull A constant level in the liquid receiver (Vl ) is a good choice with a scaledgain of 801 However according to the linear analysis the liquid level inthe condenser (Vl con) is even better with a scaled gain of 1010

bull Controlling the degree of sub-cooling in the condenser (∆Tsub= T2minusTsat(Ph))is also promising with a scaled gain of 828 but the most promising is thetemperature approach at the condenser outlet (T2minusTH) with a scaled gain of1418

bull The ratio between the implementation errorn and the optimal variation∆yopt

tells whether the implementation error or the effect of the disturbance is mostimportant for a given control policy For the most promising policies we seefrom Table44 that the contribution from the implementation error is mostimportant

Nonlinear analysis

The nonlinear model was subjected to the ldquofullrdquo disturbances to test more rigor-ously the effect of fixing alternative controlled variables The main reason for

50 Simple cycles Part IIW

s[k

W]

TH [C]10 15 20 25 30

1

2

3

4

5mdash y = Phmdash y = z- - No sub-cooling

mdash y = ∆Tsubmdash y = Vlmdash y = Vconlmdash y = T2minusTHmdash Opt

(a) Disturbance inTH (d1)

Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH

(b) Disturbance inTH (d1)

Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5

(c) Disturbance inTC (d2)

Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU

(d) Disturbance inTC (d2)

Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5

(e) Disturbance inUAloss (d3)

Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

(f) Disturbance inUAloss (d3)

Figure 43 Ammonia case Compressor power (left) and loss (right) for differentdisturbances and controlled variables A line that ends corresponds to infeasibleoperation


considering the full disturbances is to check for non-linear effects in particularpossible infeasible operation which cannot be detected from the linear analysisFigure43shows the compressor powerWs (left) and lossL =WsminusWsopt (right) fordisturbances inTH (d1) TC (d2) andUAloss (d3) Wsopt is obtained by re-optimizingthe operation for the given disturbances As predicted from the linear analysiscontrol ofPh or z should be avoided as it results in a large loss and even infeasi-bility (a line that ends corresponds to infeasible operation) Controlling the degreeof sub-cooling∆Tsub gives small losses for most disturbances but gives infeasibleoperation whenTH is low Controlling the liquid level either in the receiver orin the condenser gives small losses in all cases Another good policy is tomain-tain a constant temperature approach out of the condenser (T2minusTH) This controlpolicy was also the best in the linear analysis and has as far as we know notbeensuggested in the literature for ammonia cycles

A common design for refrigeration cycles also discussed in Part I is to have nosub-cooling in the condenser In practice this might be realized with the designin Figure41 by adding a liquid receiver after the condenser and using the chokevalve to control this liquid level or using the design in Figure411with the ldquoextrardquovalve between the condenser and tank removed The performance of thisdesign(ldquono sub-coolingrdquo) is shown with the dashed line in Figure43 The loss (rightgraphs) for this design is always nonzero as it even at the nominal point has a lossof 006kW and the loss increases with the cooling duty of the cycle Neverthelesswe note that the loss with this design is low (less than about 02kW or 35) for allconsidered disturbances This may be acceptable although it is much higherthanthe best controlled variables (Vl Vconl andT2minusTH) where the maximum losses areless than 0005kW

Figure44shows the sensitivity to implementation error for the four best controlledvariables Controlling a temperature difference at the condenser exit (either T2minusTH or ∆Tsub) has a small sensitivity to implementation error On the other handcontrolling either of the two liquid levels (Vl or Vl con) might lead to infeasibleoperation for relatively small implementation errors In both cases the infeasibilityis caused by vapour at the condenser exit In practice this vapour ldquoblow outrdquo maybe ldquofeasiblerdquo but certainly not desirable

A third important issue is the sensitivity to the total charge of the system whichis relevant for the case where we control the liquid level in the receiver (y = Vl )There is probably some uncertainty in the initial charge of the system and maybemore importantly there might be a small leak that will reduce the total charge overtime Optimally the total charge has no steady-state effect (it will only affect theliquid level in the receiver) However controlling the liquid level in the receiver(y = Vl) will make the operation depend on the total charge and we have lost one

52 Simple cycles Part IILo

ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)

Figure 44 Ammonia case Loss as function of implementation error

of the positive effects of having the liquid receiver The other control structureswill not be affected by varying total charge

Conclusion ammonia case study

For the ammonia case controlling the temperature approach at the condenser exit(T2minusTH) seems to be the best choice as the losses caused by implementation error(Figure44) and disturbances (Figure43) are very small This control implemen-tation is shown in Figure45where we also have introduced an inner ldquostabilizingrdquoloop for pressure However thesetpointfor the pressure is used as a degree offreedom so this loop does not affect the results of this study which are based onsteady-state Although not optimal even nominally another acceptable policyis

44CO2 case study 53

to use the conventional design with no sub-cooling (Figure411 with the ldquoextravalverdquo removed and minimum super-heating)

PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s

Figure 45 Proposed control structure for the ammonia cycle

44 CO2 case study

Neksaa(2002) shows thatCO2 cycles are attractive for several applications bothfrom an efficiency point of view and from an environmental perspective Skaugen(2002) gives a detailed analysis of the parameters that affect the performance of aCO2 cycle and discusses pressure control in these systems

The simple cycle studied in this paper see Figure46(a) operates between airinside a room (TC = 20C) and ambient air (TH = 30C) This could be an air-conditioner for a home as illustrated in Figure46(a) The heat loss out of thebuilding is given by Equation43 and the temperature controller shown in Figure46(a)indirectly givesQC = Qloss The nominal heat loss is 40kW

We consider a cycle with an internal heat exchanger see Figure46(a) This heatexchanger gives further cooling before the choke valve by super-heating the sat-urated vapour from the evaporator outlet This has the advantage of reducing theexpansion loss through the valve although super-heating increases thecompres-sor power For theCO2 cycle it has been found that the internal heat exchangerimproves efficiency for some operating points (Domanski et al 1994) For the


CO2 cycle we find that the internal heat exchanger gives a nominal reduction of99 in Ws For the ammonia cycle the effect of internal heat exchange to givesuper-heating is always negative in terms of efficiency

441 Modelling

Table41shows the structure of the model equations and the data are given in Table45 Constant air temperature is assumed in the evaporator (TC) The gas coolerand internal heat exchanger are modelled as counter-current heat exchangers with6 control volumes each The Span-Wagner equation of state (1996) is used forthe thermodynamic calculations The MATLAB model is available on the internet(Jensen and Skogestad 2007a)

z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1


Figure 46 TheCO2 cycle operates trans-critical and is designed with an internalheat exchanger

442 Optimal operation

Some key parameters for optimal operation of theCO2 cycle are summarized inTable46 and the pressure enthalpy diagram is given in Figure46(b) Figure47shows the optimal temperature profiles in the gas cooler and in the internal heatexchanger

44CO2 case study 55

Table 45 Conditions for theCO2 case studyEvaporator(UA)vap = 798WC-1

Gas cooler(UA)gco = 795WC-1

Internal heat exchanger(UA)ihx = 153WC-1

Compressor isentropic efficiencyη = 075Ambient TH = 30CAir flow gas coolermcp = 250JC-1s-1

RoomTC = TsC = 20C

RoomUAloss= 400WC-1

Choke valveCV = 121middot10minus6m2

Table 46 Optimal operation forCO2 caseWs[W] 958

z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312

Note that when the ambient air goes below approximatelyTH = 25C the opti-mal pressure in the gas cooler is sub-critical We will only consider trans-criticaloperation so we assume that the air-conditioner is not used below 25C


We want to find what the valve should control In addition to the variables listedinSection42 we also consider internal temperature measurements in the gas coolerand internal heat exchanger Note that the ldquono sub-coolingrdquo policy is not possiblefor theCO2 cycle because it operates trans-critical

As discussed in more detail below there are no obvious single measurementstocontrol for this application One exception is the holdupm on the high pressureside of the cycle However measuring the holdup of a super-critical fluidis not


Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4

(b) Internal heat exchanger

Figure 47 CO2 case Temperature profile in gas cooler and internal heat ex-changer

easy (one might use some kind of scale but this will be to expensive in most ap-plications) Thus we will consider measurement combinations First we will tryto combine two measurements and if this is not acceptable for all disturbanceswe may try more measurements Any two measurements can be combined andwe choose here to combinePh and T2 The reason is thatPh is normally con-trolled anyway for dynamic reasons andT2 is simple to measure and is promisingfrom the linear analysis Also temperature corrected setpoint for pressure hasbeen proposed before (Kim et al 2004) We use the ldquoexact local methodrdquo (Al-stad and Skogestad 2007) and minimize the 2-norm ofMd = HFWd whereF =partyoptpartdi is the optimal sensitivity ofyprime = [Ph T2] with respect to disturbancesdprime = [TH TC (UA)loss] The magnitude of the disturbances are given inWd Wefind that the linear combinationc= h1 middotPh+h2 middotT2 with k= h2h1 =minus853barC-1

minimizes the 2-norm of the three disturbances on the loss This can be imple-mented in practice by controlling the combined pressure and temperature

Phcombine= Ph +k middot(T2minusT2opt

)(45)

whereT2opt = 255C andk = minus853barC-1 An alternative is to use a morephysically-based combination For an ideal gas we havem = PVmiddotMW

RT and sincethe gas cooler holdupmgco seems to be a good variable to control we will includePT in the gas cooler as a candidate controlled variable

44CO2 case study 57

Table 47 Linear ldquomaximum gainrdquo analysis of controlled variables forCO2 case|∆yopt(di)|

Variable (y) Nom G d1 (TH ) d2 (TC) d3 (UAloss) |∆yopt| n spany |Gprime|

PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime

2 minusTH [C] 362 24 410 -192 500 675 15 825 291z[-] 034 1 015 -004 018 024 005 029 345Vl [m3] 007 003 -002 0005 -003 0006 0001 0007 477T2 [C] 255 6014 837 090 318 900 1 100 602Phcombine[bar] 9761 -5920 -231 -231 391 330 953 425 139mgco[kg] 483 -1118 0151 -0136 0119 0235 044 0675 1655

Linear method

We first use the linear ldquomaximum gainrdquo method to find promising controlled vari-ables The following disturbanceslowast are considered


d2 ∆TC = plusmn5C

d3 ∆UAloss from minus100 to +40WC-1

The linear results are summarized in Table47 Some controlled variables (Pl T prime4

∆Tsub and∆Tsup) are not considered because they as discussed earlier can not befixed or are not relevant for this cycle The ratioPhT prime

2 in the gas cooler is notfavourable with a small scaled gain This is probably because the fluid in thegascooler is far from ideal gas soPhT prime

2 is not a good estimate of the holdupmgcoFrom Table47 the most promising controlled variables are the holdup in the gascooler (mgco) and the linear combination (Phcombine) Fixing the valve openingzs

(no control) or the liquid level in the receiver (Vl ) are also quite good

Non-linear analysis

Figure48 shows the compressor power (left) and loss (right) for some selectedcontrolled variables We see that the two most important disturbances are thetem-peraturesTH andTC which gives larger losses than disturbance in the heat loss outof the building Controlling the pressurePh gives infeasible operation for small



s[k

W]

TH [C]25 30 35 400

1

2

3

4

5mdash y = Phmdash y = T2mdash y = z- - y = Vl vap- - y = mgco y = Phcombinemdash Opt

(a) Disturbances inTH (d1)

TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph


Figure 48CO2 case Compressor power (left) and loss (right) for different dis-turbances and controlled variables A line that ends corresponds to infeasible op-eration

44CO2 case study 59

disturbances in the ambient air temperature (TH) The nonlinear results confirmthe linear gain analysis with small losses forPhcombineandmgco

Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)

Figure 49CO2 case Loss as function of implementation error

Another important issue is the sensitivity to implementation error From Figure49we see that the sensitivity to implementation error is very large fory = Vl Thethree best controlled variables are constant valve opening (z) constant holdup inthe gas cooler (mgco) and the linear combination (Phcombine)

ConclusionCO2 case study

For thisCO2 refrigeration cycle we find that fixing the holdup in the gas coolermgco gives close to optimal operation However since the fluid is super-critical


holdup is not easily measured Thus in practice the bestsinglemeasurement is aconstant valve openingz (ldquono controlrdquo) A better alternative is to usecombinationsof measurements We obtained the combinationPhcombine= Ph + k middot (T2minusT2opt)using the ldquoexact local methodrdquo This implementation is shown in Figure410 Thedisturbance loss compared with single measurements is significantly reduced andthe sensitivity to implementation error is very small

z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4

Figure 410 Proposed control structure for theCO2 cycle

45 Discussion

451 Super-heating

An important practical requirement is that the material entering the compressormust be vapour (either saturated or super-heated) Saturation can be achieved byhaving a liquid receiver before the compressor as shown in Figure41 Howeverin many designs the receiver is located at the high pressure side and super-heatingmay be controlled with the choke valve (eg thermostatic expansion valve TEV)as shown in Figure411 A minimum degree of super-heating is required to handledisturbances and measurement errors Since super-heating is not thermodynami-cally efficient (except for some cases with internal heat exchange) this minimaldegree of super-heating becomes an active constraint With the configuration inFigure 411 the ldquoextrardquo valve is the unconstrained degree of freedom (u) that

45 Discussion 61

should be adjusted to achieve optimal operation Otherwise the results from thestudy hold both for the ammonia andCO2 cycle

TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl

ldquoExtrardquovalve

Figure 411 Alternative refrigeration cycle with liquid receiver on high pressureside and control of super-heating

452 Heat transfer coefficients

We have assumed constant heat transfer coefficients in the heat exchangers Nor-mally the heat transfer coefficient will depend on several variables such as phasefraction velocity of the fluid and heat transfer rate However a sensitivity analysis(not included) shows that changing the heat transfer coefficients does not affectthe conclusions in this paper For theCO2 cycle we did some simulations using aconstant air temperature in the gas cooler which may represent a cross flow heatexchanger and is an indirect way of changing the effective UA value We found thatthe losses for a constant liquid level control policy (y = Vl ) was slightly smallerbut the analysis presented here is still valid and the conclusion that a combinationof measurements is necessary to give acceptable performance remains the same

453 Pressure control

This paper has only considered steady-state operation For dynamic reasons inorder to ldquostabilizerdquo the operation a degree of freedom is often used to control onepressure (Pl or Ph) However thesetpointfor the pressure may be used as a degreeof freedom at steady-state so this will not change the results of this study An

62 Bibliography

example of a practical implementation using cascade control is shown in Figure45 where the temperature difference at the condenser outlet is controlled whichwas found to be the best policy for the ammonia case study The load in the cycleis controlled by adjusting the setpoint to the pressure controller that stabilize thelow pressure (Pl )

46 Conclusion

For a simple cycle there is one unconstrained degree of freedom that should beused to optimize the operation For the sub-critical ammonia refrigeration cycleagood policy is to have no sub-cooling Further savings at about 2 are obtainedwith some sub-cooling where a good control strategy is to fix the temperatureapproach at the condenser exit (T2minusTH) see Figure45 One may argue that 2savings is very little for all the effort but larger savings are expected for caseswith smaller heat exchanger areas (Jensen and Skogestad 2007b) and allowingfor sub-cooling shows that there is no fundamental difference with theCO2 case

For the trans-criticalCO2 cycle the only single ldquoself-optimizingrdquo measurementseems to be the holdup in the super-critical gas cooler (mgco) However sincethis holdup is difficult to measure a combination of measurements is needed Wepropose to fix a linear combination of pressure and temperaturePhcombine= Ph +k middot (T2minusT2opt) see Figure410 This is a ldquoself-optimizingrdquo control structure withsmall losses for expected disturbances and implementation errors

Acknowledgment

The contributions of Tore Haug-Warberg and Ingrid Kristine Wold on implement-ing the thermodynamic models are gratefully acknowledged

Bibliography

Alstad V and Skogestad S (2007) lsquoThe null space method for selecting optimalmeasurement combinations as controlled variablesrsquoInd Eng Chem Res

Domanski P A Didion D A and Doyle J P (1994) lsquoEvaluation of suction-lineliquid-line heat exchange in the refrigeration cyclersquoInt J Refrig17 487ndash493

Bibliography 63


Halvorsen I J Skogestad S Morud J C and Alstad V (2003)lsquoOptimal selec-tion of controlled variablesrsquoInd Eng Chem Res42 3273ndash3284

Jensen J B and Skogestad S (2007a) gPROMS and MATLAB model code forammonia andCO2 cycles See additional material for paper at homepage of SSkogestad

Jensen J B and Skogestad S (2007b) lsquoOptimal operation of simple refrigera-tion cycles Part I Degrees of freedom and optimality of sub-coolingrsquoComputChem Eng31 712ndash721




Neksaa P (2002) lsquoCO2 heat pump systemsrsquoInt J Refrigeration25 421ndash427

Skaugen G (2002) Investigation of TranscriticalCO2 Vapour Compression Sys-tems by Simulation and Laboratory Experiments PhD thesis Norwegian Uni-versity of Science and Technology


Skogestad S and Postlethwaite I (2005)Multivariable feedback control Secondedn John Wiley amp Sons

Span R and Wagner W (1996) lsquoA new equation of state for carbondioxidecovering the fluid region from the triple-point temperature to 1100 K at pressuresup to 800 MParsquoJ Phys Chem Ref Data25(6) 1509ndash1596



64 Bibliography

Chapter 5

Problems with specifying∆Tmin indesign of processes with heatexchangers

Accepted for publication in Industrial amp Engineering Chemistry Research

We show in this paper that the common method of specifying∆Tmin

for individual heat exchangers may lead to wrong decisions and shouldbe used with care when designing heat exchanger systems In particu-lar design with constraints on∆Tmin may result in operation conditionswhich are not optimal when the resulting areas are installed In additiondifferentU-values for the heat exchangers are not easily handled Wepropose an alternative method (simplified TAC) to avoid these problemsand compare it with the∆Tmin-method on three vapour compression (re-frigeration) cycle case studies

51 Introduction

In process design one seeks to optimize the future income of the plant Thismight be realized by minimizing the total annualized cost (TAC)JTAC = Joperation+Jcapital[$year-1] see Problem51 below However findingJTAC requires detailedequipment and cost data which is not available at an early design stage

An alternative simple and common approach for design of processes with heat ex-changers especially at an early design stage is to specify the exchanger minimumapproach temperature (EMAT= ∆Tmin) in each heat exchanger see Problem52

65

66 Problems with the minimum approach temperature

below The idea is that this specification should give a reasonable balancebetweenminimizing operating costsJoperation (favored by a small∆Tmin) and minimizingcapital costsJcapital (favored by a large∆Tmin)

As an example Figure51 shows a hot stream (T2) transferring heat to a coldstream with constant temperature (T1) Stream 2 may be hot exhaust gas which iscooled to recover its energy and this is done by vaporizing water in stream 1 Asmall value of∆Tmin means that a lot of the energy is recovered but it requiresa large heat exchanger On the other hand a larger value of∆Tmin requires lessarea but the outlet temperatureT2 will be higher and less energy is recoveredThere exists many rules of thumb for the value of∆Tmin For exampleTurton et al(1998 page 250) recommends 10C for fluids and 5C for refrigerants

0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)

Figure 51 The effect of different values for∆Tmin

Note that what we call∆Tmin in this paper is the individual exchanger minimumapproach temperature (EMAT) This should not be confused with the ldquopinch tem-peraturerdquo (heat recovery approach temperature HRAT) used in design of heat ex-changer networks

The use of∆Tmin as a constraint is only used fordesign as it is reasonably wellknown that we should never specify∆Tmin during operation (here the areas shouldbe used as constraints rather than∆Tmin) However even when it comes to designspecifying∆Tmin (and maybe varying it in an outer loop) does not always resultin a good design except for simple cases To understand why specifying ∆Tmin indesign may not be correct consider the following three problems

Problem 51 Detailed optimal designbased on minimizing TAC[$year-1] (egBiegler et al 1997)

minu1

(Joperation+Jcapital

)(51)

51 Introduction 67

subject to model equations and operational constraints (this applies to all problemsin this paper) The degrees of freedomu1 include all the equipment data (sizes)and operating variables The annualized capital cost is often obtained as

Jcapital= sumiisinUnits

(Cfixedi+CvariableimiddotSnii )T (52)

HereSi is the characteristic size for the unit (area in m2 for heat exchangers) andthe cost factors (Cfixedi andCvariablei) and cost scaling factorni are constants foreach unit (eg heat exchangers)T is the capital depreciation time egT =10years The operating costJoperationare given by the prices of feeds products andutilities (energy) plus other fixed and variable operating costs eg see Equation58below

Problem 52 Simplified optimal designwith specified∆Tmin

minu2

(Joperation) (53)

subject to ∆Ti minus∆Tmin ge 0

Here the degrees of freedomu2 include the heat transfer in the heat exchanger(Qi) and the operating variables (flows works splits etc) After solving this prob-lem one can calculate the heat exchanger areasAi from the resulting temperatures(usingQi =

intUi∆Ti dAi) Note that Problem52will favor designs where the tem-

perature difference∆T is close to∆Tmin throughout the heat exchangers becausethis improves energy efficiency but does not cost anything Specifying∆Tmin willtherefore tend to give designs with large heat exchanger areas In addition dif-ferentU-values can not be handled easily as they are not part of the optimizationproblem in Equation53 An indirect approach is too use different∆Tmin i for eachheat exchanger in Equation53

Let us now consider steady-state operation where the equipment data includingheat exchanger areas are given and the degrees of freedomu3 include only theoperating variables For each heat exchanger there is at steady-state one operat-ing variable which may be chosen as theeffectiveareaAi (in practiceAi may bechanged using a bypass) However we must requireAi le Amax

i whereAmaxi is the

ldquoinstalled areardquo eg found from Problem51or 52 We then have

Problem 53 Optimal operationwith given heat exchanger areas

minu3

(Joperation) (54)

subject to Ai minusAmaxi le 0


Note that in many cases including the examples in this paper it is optimal to haveAi = Amaxi

The solution to Problem53in terms of optimal stream data (temperatures) will bethe same as to Problem51 butnot generally the same as to Problem52 see themotivating example below To understand this note that in Problem53 with theareas given there is no particular incentive to make the temperature difference∆Tldquoevenrdquo (due to∆Tmin) throughout the heat exchangers Provided there are degreesof freedom we will therefore find that∆T from Problem53varies more throughthe heat exchangers than∆T from Problem52 In particular the∆Tmin obtainedfrom Problem53 is often smaller than that specified in design (Problem52)see the introductory example below Thus the optimal nominal operating point(solution to Problem53) is not the same as the nominal simplified design point(solution to Problem52) From this it is clear that specifying∆Tmin in design isnot a good approach

The objective of this paper is to study the∆Tmin-method (Problem52) in moredetail and suggest an alternative simple design method (called the simplified TAC-method) for heat exchanger systems The optimization problems are solved usingthe gPROMS software

TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl

Figure 52 An ammonia refrigeration system

52 Motivating example Ammonia refrigeration cycle

The ammonia refrigeration cycle for cold storage presented inJensen and Skoges-tad(2007a) is shown in Figure52 We use the following conditions

52 Motivating example Ammonia refrigeration cycle 69

Table 51 Structure of model equations for the ammonia case studyHeat exchangers (condenser and evaporator)Q =

intU∆T dA = mmiddot (houtminushin)




bull Qloss= 20kW


bull Cold storage (indoor) temperature set pointTsC = minus12C

bull Heat transfer coefficient for the evaporator and condenserU = 500Wm-2 C-1

bull The ammonia leaving the evaporator is saturated vapour (which is alwaysoptimal for this cycleJensen and Skogestad(2007b))

The temperature controller is assumed to adjust the compressor power to maintainTC = Ts

C which indirectly sets the loadQC = Qloss The main model equations aregiven in Table51

521 ∆Tmin design-method

The operational cost is given by the compressor power (Joperation=Ws) so with the∆Tmin-method the optimal design problem see Problem52 becomes

minu2

(Ws)

subject to ∆Tvapminus∆Tmin vapge 0 (55)

∆Tconminus∆Tminconge 0

where the degrees of freedomu2 include the heat transfered in the two heat ex-changers (Qi) (but note thatQC = 20kW) plus three other operating variableslowast

(eg two pressures and refrigerant flow) We choose∆Tmin = 10C in both the

lowastThe simple cycle in Figure52has five operating variables (Jensen and Skogestad 2007b) buttwo of these have been specified (given load and saturated vapour)


evaporator and the condenser (Shelton and Grossmann 1986) The resulting heatexchanger areasA are then obtained fromQ =

int(U middot∆T)dA

As noted in the introduction the solution to Equation55 does not generally giveoptimal operation with the resulting areas The re-optimized operation problemwith given areas see Problem53 becomes

minu3

(Ws)

subject to AvapminusAmaxvap le 0 (56)

AconminusAmaxcon le 0

whereAmaxvap andAmax

con are the result of the∆Tmin-method design problem (Equation55) The degrees of freedomu3 include Avap and Acon (which in this case areoptimally equal to their maximum design values) plus the three other operatingvariables (eg two pressures and refrigerant flow)

The results for the two problems are summarized in the two left columns of Table52 and we note that re-optimization reduces the operating cost (Ws) by 32Figure53shows the corresponding temperature profiles in the condenser

Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120

(a) Optimal design with specified∆Tmin

Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120

(b) Re-optimized operation with specifiedA

Figure 53 Temperature profile in the condenser for the∆Tmin-method

bull In the design case (55) (with fixed∆Tmin) there is no sub-cooling of ammo-nia in the condenser In the re-optimized operation (56) however there is asub-cooling of 89C The optimality of sub-cooling in simple refrigerationcycles is discussed in detail inJensen and Skogestad(2007b)

bull The high pressurePh is increased by 07 in the re-optimized case but thisis more than compensated for by a 37 reduction in flowrate

53 Proposed simplified TAC method 71

Table 52 Ammonia case study∆Tmin-method Simplified TAC (Eq59)

Design (55) Operation (56) C0 = 818lowast C0 = 8250dagger

∆Tmin = 10C Re-optimized∆Tvap

min [C] 100 100 913 267∆Tcon

min [C] 100 153 184 100Acon[m2] 450 450 412 147Avap[m2] 400 400 438 150Atot [m2] 850 850 850 297HX cost[-] 100 100 100 051Pl [bar] 174 174 181 077Ph [bar] 135 136 140 284∆Tsub[

C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160

Given data is shown in boldfacelowastC0 adjusted to get same total heat exchanger area as the∆Tmin-methoddaggerC0 adjusted to get same∆Tmin as used in the∆Tmin-method

In summary we find that the∆Tmin design-method does not result in optimal oper-ating variables for the given example A better design method is therefore needed

53 Proposed simplified TAC method

The original cost function for Problem51 requires quite detailed cost data plusa lot of other information which is not available at an early design stage Thesimplified Problem52on the other hand is easy to formulate and solve but herewe cannot easily handle differentU-values for heat exchangers and the optimaldesign point is not generally the same as the optimal operating point (even nom-inally) Therefore the objective is to find a better simplified formulation Thestarting point is to replace the equipment cost (Equation52) in Problem51witha simplified expression First we assume that the structure of the design is givensuch that we need not consider the fixed cost terms (ie we setCfixedi = 0) Sec-ond we only consider heat exchanger costs For a vapour compression cycle thisis justified if the capital cost for the compressor is proportional to the compressorpowerWs which corresponds to assumingni = 1 for the cost exponent for the com-pressor We can then include the capital cost for the compressor in the operatingcost of the compressor Third we assume that all heat exchangers have the same


cost factors (Cvariablei= C0 andni = n)

The resulting ldquosimplified TACrdquo optimal design problem becomes

Problem 54 Simplified optimal designwith simplified TAC

minu4

(

Joperation+C0 middotsumi

Ani

)

(57)

subject to model equations and operating constraints and whereu4 includes theheat exchanger areasAi plus the operating variables In the general case

Joperation= sum pFi Fi minussum pPj Pj +sum pQkQk +sum pWslWsl [$year-1] (58)

whereFi are feedsPj are productsQk are utilities (energy)Wsl are the mechan-ical work the prsquos are respective prices For a heat exchanger network problemthis can be reduced toJoperation= sum pQi Qi [$year-1] which in many cases simpli-fies to Joperation= QH [$year-1] whereQH is the supplied heat (Gundersen andNaess 1988) For the refrigeration cycles considered in this paperJoperation=Ws[$year-1]

In the examples we choosen = 065 for the heat exchangers and useC0 as thesingle adjustable parameter (to replace∆Tmin) There are several benefits comparedwith the∆Tmin-method

bull The heat exchanger temperatures depend onUiAi for each heat exchangerThus differentU-values for each exchanger are easily included

bull The optimal design (Equation57) and the optimal operation (Problem53)have the same solution in terms of optimal stream data This follows sincethe termC0 sumi A

ni is constant in operation

bull The assumption of using the sameC0 for all heat exchangers is generally amuch better than assuming the same∆Tmin

On the other hand compared to the∆Tmin design method the simplified TAC de-sign method requires calculation of∆T inside all exchangers during the optimiza-tion and the optimization problem is also a bit more difficult to solve Howeverthe proposed method does not require any additional data compared with the∆Tmin

method

54 Other case studies 73

531 Revisit of ammonia case study

The optimization problem (Equation57) for the proposed simplified TAC-methodbecomeslowast

min(Ws+C0 middot (Ancon+An

vap)) (59)

The right two columns of Table52shows the optimal design withn= 065 and twodifferent values ofC0 FirstC0 = 818 gives the same total heat exchanger area andalmost the same capital cost as the∆Tmin-method but the area is better distributedbetween the evaporator and condenser This results in a 360 reduction in op-erating cost (Ws) compared with the∆Tmin-method (036 after re-optimizing theoperating point for the∆Tmin-method) SecondC0 = 8250 gives∆Tmin = 100Cas specified in the∆Tmin method The compressor work is increased with 107(114) but the heat exchanger area is reduced by 60 and this is theonly designthat truly satisfies the∆Tmin we selected initially

The simplified TAC method confirms that sub-cooling is optimal and we see thatthe degree of sub-cooling increases with decreasing heat transfer area (increasedC0)

Note that the heat transfer coefficientsUi were assumed to be equal but the sim-plified TAC method will automatically distribute the heat transfer area optimallyalso if the heat exchangers have different heat transfer coefficients For examplewith Uvap = 2Ucon the energy savings (for the same heat exchangers cost) are evenlarger (6) using the simplified TAC method compared with the∆Tmin-method

54 Other case studies

We here briefly present results from two other case studies

541 CO2 air-conditioner

CO2 as a working fluid in air-conditioners and heat-pumps is gaining increasedpopularity because of its low environmental impact (Lorentzen 1995 Neksaa2002) We consider a trans-criticalCO2 air-condition unit with the following data

lowastIn a more realistic design one may also consider additional constraints such as maximumcompressor suction volumes and pressure ratio but this is not discussed here


Table 53CO2 air-conditioner∆Tmin-method Simplified TAC

Design Operation C0 = 253lowast C0 = 185dagger C0 = 877Dagger

∆Tmin = 5C Re-optimized∆Tgco

min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx

min [C] 500 475 - - -Agco[m2] 131 131 176 202 092Avap[m2] 160 160 138 160 070Aihx [m2] 023 023 0 0 0Atot [m2] 314 314 314 362 161HX cost[-] 100 100 091 100 059Ph [bar] 878 916 928 910 1070Pl [bar] 508 508 499 508 433m[mols-1] 065 059 069 070 067Ws[W] 892 859 871 814 1328COP[-] 449 465 459 492 301

Given data is shown in boldfacelowastC0 adjusted to get same total area as∆Tmin-methoddaggerC0 adjusted to get same heat exchanger cost as the∆Tmin-methodDaggerC0 adjusted to get same∆Tmin as used in the∆Tmin-method

bull Heat transfer coefficientU = 500Wm-2K-1 for the evaporator condenserand internal heat exchanger


bull Set point for room temperatureTC = 20C

bull Heat loss into the roomQloss= 40kW

The details about the model are found inJensen and Skogestad(2007a) In the op-timization we have included an internal heat exchanger (with areaAihx) that trans-fers heat from before the compressor to before the valve Otherwise the flowsheetis as for the ammonia cycle shown in Figure52

For solving Problem52 we use a design∆Tmin = 50C in all heat exchangersAgain we find that re-optimizing for operation (Problem53) gives a better oper-ating point with 370 less compressor power The results given in Table53aresimilar to the ammonia cooling cycle although there is no sub-cooling sincePh isabove the critical pressure

Interestingly with the simplified TAC method we obtainAihx = 00m2 whichmeans that it is not optimal from an economical point of view to pay for the area for


the internal heat exchanger (although the internal heat exchanger would of coursebe used if it were available free of charge) This is a bit surprising sincewe havenot included the fixed cost of installing a heat exchanger which would make iteven less desirable to invest in an internal heat exchanger On the otherhand ifwe require a lot of super-heating before the compressor then it might be better toachieve this super-heating in an internal heat exchanger but this is notdiscussedhere

With C0 = 253 we get the same total heat transfer area as for the∆Tmin-method butthe shaft work is reduced by 426 (058 compared to re-optimized)C0 = 185gives the same cost of heat exchanger area (without even considering the savingsof completely removing a heat exchanger) andWs is reduced by 1222 (885)With C0 = 877 we get the only design with∆Tmin = 50C The heat exchangercost is reduced by 41 and the compressor power is increased by 49 (55)compared with the∆Tmin-method

542 PRICO LNG process

The PRICO LNG process (Price and Mortko 1996) is a simple configuration uti-lizing mixed refrigerants Details about the model is presented elsewhere (Jensen2008) Note that we are not considering constraints on compressor suction volumeand pressure ratio for the compressor This will be important in an actual designbut we have tried to keep the case study simple to illustrate the effect of specifying∆Tmin

A design∆Tmin of 20C is used for the∆Tmin method From Table54 we seethat re-optimizing reduces the energy usage (Ws) by 48 This is achieved byincreasing the pressure ratio (by 255) and reducing the refrigerant flowrate (by167) The composition of the refrigerant is also slightly changed but this isnot shown in Table54 We were quite surprised by the rather large improvementobtained by re-optimizing with fixed heat transfer areas considering the relativelylow value for the initial∆Tmin

With the simplified TAC method we get a 41 reduction (02 increase com-pared to re-optimized) inWs for the same total heat transfer area (C0 = 2135) Thesmall increase inWs compared with the re-optimized∆Tmin design is because thesimplified TAC method minimizes the heat exchanger cost and not the total areaWith the same cost (C0 = 2090) the TAC-method gives a reduction in compressorpower of 43 (01) The saving compared with the re-optimized case is smallbecause of the small∆Tmin resulting in very large heat exchangers A more rea-sonable design is achieved withC0 = 7350 which gives a design with a true∆Tmin


Table 54 PRICO LNG process∆Tmin-method Simplified TAC

Design Operation C0 = C0 = C0 =∆Tmin = 2C Re-optimized 2135lowast 2090dagger 7350Dagger

∆TminHOT [C] 200 089 090 086 200∆TminNG[C] 200 098 109 108 222AHOT middot10minus3 [m2] 982 982 1012 1027 431ANG middot10minus3 [m2] 299 299 269 272 145ATot middot10minus3 [m2] 1281 1281 1281 1299 577HX cost[-] 100 100 099 100 060Ph [bar] 201 271 270 268 378Pl [bar] 27 29 29 29 191m[kmols-1] 30 25 25 25 23Ws[MW] 1894 1814 1817 1812 2216

Given data is shown in boldfacelowastC0 adjusted to get same total area as∆Tmin-methoddaggerC0 adjusted to get same heat exchanger cost as the∆Tmin-methodDaggerC0adjusted to get same∆Tmin as used in the∆Tmin-method

of 20C The heat exchanger capital cost is reduced by 40 but the compressorpower is increased by 170 (222)

55 Discussion

551 ∆Tmin-method

There are some main points that are important to note from this analysis of the∆Tmin-method

1 ∆Tmin is treated as an important parameter in heat exchanger design butthe theoretical basis seems weak as ldquoviolatingrdquo∆Tmin in operation may givelower operating cost

2 The∆Tmin-method will not always give the optimal operating point so sub-optimal setpoints might be implemented

3 The size distribution between the heat exchanger will not be optimal al-though this may be partly corrected for by individually adjusting the valueof ∆Tmin for each heat exchanger

4 More seriously the results might lead to wrong structural decisions andthis

55 Discussion 77

can not be changed by iterating on the∆Tmin-values In the ammonia casestudy one would incorrectly conclude that sub-cooling is not optimal andthus implement a liquid receiver after the condenser This would duringoperation achieve no sub-cooling From the true optimum however we seethat some sub-cooling is optimal

5 One potential advantage with the∆Tmin-method is that it only requires anoverall energy balance for the heat exchangers However for more com-plex cases a more detailed model of the heat exchangers is needed so in thegeneral cases this advantage is lost

In summary the∆Tmin-method is not satisfactory for realistic design problems

552 Other approaches

The question is whether there are other ways of specifying temperature differencesIn terms of area rather than specifying∆Tmin it would be better to specify themean temperature difference∆T = 1

A

int∆TdA since∆T is directly linked to the

heat transfer area by (assuming constant heat transfer coefficient)

Q = UA∆T (510)

However this method has several drawbacks that makes it unattractive touseFirst it might be cumbersome to calculate the integral of the temperature and sec-ond and more importantly there are no general rules in selecting values for∆TWe therefore propose to use the simplified TAC-method

553 Heat exchanger network design

The results in this paper show that one should notuse a constraint on∆Tmin (EMAT)for the design of individual heat exchangers What about the standard heat ex-changer network (HEN) design problem (egGundersen and Naess 1988) wherea constraint on the heat recovery approach temperature∆Tmin (HRAT) for the net-work is used Our results do notinvalidate this approach First the stream data(inlet and outlet temperatures and flows) are fixed for the standard HEN prob-lem It then follows that specifying∆Tmin (HRAT) is equivalent to specifying theheat recovery or equivalently the required hot utility (QH) The solution to thisparticular design problem will therefore result in optimal operating data if wein-stall the resulting areas (and remove the∆Tmin specification) However note thatthe simplified TAC-formulation in Equation57with Joperation= sumi pQi Qi is much

78 Bibliography

more general than the standard heat exchanger network design problem where thestream data are specified

56 Conclusion

We have shown that the method of specifying∆Tmin for design of heat exchangers(min J subject to∆T ge ∆Tmin) may fail to give an optimal operating point In theammonia refrigeration case study the∆Tmin-method fails to find that sub-coolingin the condenser is optimal As a simple alternative we propose the simplified totalannualized cost (TAC) method (min(J+C0 sumi A

ni )) whereC0 replaces∆Tmin as the

adjustable parameter A high value ofC0 corresponds to increasing the investment(capital) costs relative to the operating (energy) costs and favors small areas and alarger∆Tmin ThusC0 can be adjusted to get a desired value for∆Tmin or the totalarea or it can be obtained from cost data With the alternative method differentheat transfer coefficientsUi can also be accounted for

Another important conclusion is related to the temperature difference profilein theheat exchanger According to exergy or entropy minimization rules of thumb(egSauer et al 1996) it is optimal to have even driving forces which suggests that∆Tshould be constant in heat exchangers The results presented here however suggestthat this is not true The∆Tmin approach (Problem52) favors a more constant∆Tprofile (see Figure53(a)) but in optimal operation (Problem53) we find that thetemperature difference is small in one end (see Figure53(b))

Bibliography

Biegler L T Grossmann I E and Westerberg A W (1997)Systematic Methodsof Chemical Process Design Prentice Hall

Gundersen T and Naess L (1988) lsquoThe synthesis of cost optimalheat exchang-ers networksrsquoComput Chem Eng12(6) 503ndash530

Jensen J B (2008) Optimal operation of cyclic processes With application toLNG processes PhD thesis Norwegian University of Science and Technology

Jensen J B and Skogestad S (2007a) lsquoOptimal operation of a simple refrig-eration cycles Part II Selection of controlled variablesrsquoComput Chem Eng31 1590ndash1601

Bibliography 79


Lorentzen G (1995) lsquoThe use of natural refrigerants a complete solution to theCFCHCFC predicamentrsquoInt Journal of Refrigeration18 190ndash197


Price B C and Mortko R A (1996) PRICO - a simple flexible provenapproachto natural gas liquefactionin lsquoGASTECH LNG Natural Gas LPG interna-tional conference Viennarsquo

Sauer E Ratkje S K and Lien K M (1996) lsquoEquipartition of forces A newprinciple for process design and optimizationrsquoInd Eng Chem Res35 4147ndash4153

Shelton M R and Grossmann I E (1986) lsquoOptimal synthesis of integrated re-frigeration systemsmdashi Mixed-integer programming modelrsquoComput ChemEng10(5) 445ndash459

Turton R Bailie R C Whiting W B and Shaeiwitz J A (1998)Analysissynthesis and design of chemical processes Pearson Education Ltd

80 Bibliography

Chapter 6

Optimal design of a simple LNGprocess

In this chapter we discuss the design optimization of a single cyclemixed fluid LNG process the PRICO process A simple objective func-tion is stated and used on nine different cases with varying constraintsImportant constraints are discussed and the results are compared withthe commercial PRICO process and with other publications

61 Introduction

Large amounts of natural gas are found at locations that makes it infeasible or noteconomical to transport it in gaseous state to the customers The most economicway of transporting natural gas over long distances is to first produce liquefiednatural gas (LNG) and then transport the LNG by ships LNG has approximately600 times the density of gaseous natural gas

At atmospheric pressure LNG has at saturated conditions a temperature of ap-proximatelyminus162C so the process requires large amounts of energy to generatethe necessary cooling Several different process designs are used and they can begrouped roughly as follows

1 Pure fluid cascade process The refrigerants are pure fluids but several cy-cles are used to improve the efficiency

2 Mixed fluid refrigerant The refrigerant composition is adjusted to matchthe cooling curve of the natural gas

81

82 Optimal design of a simple LNG process

3 Mixed fluid cascade process Combination of the two where energy effi-ciency is further improved by using several mixed refrigerant cycles

The PRICO LNG process considered in this paper see Figure61 belongs to thesecond group and has a single cycle with a mixed refrigerant The mixed refrig-erant gives a good temperature match throughout the heat exchanger comparedwith using pure component cycles (group 1) However compared with multiplemixed refrigerant processes the thermodynamic efficiency is lower so theprocessis mainly used for smaller plants (eg peak shaving plants) up to 2MTPA (milliontons per annum)

Stebbing and OrsquoBrien(1975) reports on the performance of the first commercialPRICO plants in operationPrice and Mortko(1996) from the Black amp Veatchcompany discuss the process and give some key values for several oftheir plantsWith respect to academic workLee et al(2002) used the PRICO process as oneof their case studies for testing their approach to design optimization The samegroup later published some updated results (Del Nogal et al 2005)

LNG Flash gas

Fuel gasPre-treatedNG SW

Fuel HXNG HX

Refrigerantcompressor

Fuelcompressor

Condenser Ph

Pl

PmTout

Figure 61 Simplified flowsheet of the PRICO process also showing flash gasheating and recompression (not considered here)

61 Introduction 83

611 Optimal design

Mathematically optimal design may be expressed as the solution to the followingoptimization problem

min JTAC = Joperation+Jcapital (61)

subject to cle 0

whereJcapital[$year-1] is the annualized cost of the equipment andJoperation[$year-1]is the annual operating cost (Joperation= Jutility +Jfeeds+Jproducts) The total annual-ized cost (TAC) is minimized with respect to the design variables which includesboth the structure (integer variables) and the design parametersc is a set of designconstraints (ie maximum design pressure non-negative flows and feedcomposi-tion)

The truly optimal design is a complex task involving mixed integer non-linearprogramming In our case however we start from a given process structure so wemay use a simpler approach with no integer variables

For heat exchanger design a common approach is to minimize only the operatingcost but subject to a minimum approach temperature in all heat exchangers(Leeet al 2002 Del Nogal et al 2005) that is∆Ti ge ∆Tmin is introduced as a designconstraint The idea is that the heat exchanger minimum approach temperaturelowast∆Tmin gives a balance between low operating cost (favored by low∆Tmin) and lowcapital cost (favored by high∆Tmin) This approach will however not generallyresult in a true optimal design even when iterating on∆Tmin Specifically re-optimizing to find the optimal operation given the resulting equipment will usuallyresult in a lower optimal value of∆Tmin (Chapter5) This also implies that a designapproach based on specifying∆Tmin and altering it in an outer loop will not work

Therefore we choose to use the simplified total annual cost (sTAC) methodpre-sented in Chapter5 The capital cost is often expressed as a fixed and a variablecontributionJcapital= sumi (Cfixedi+CvariableimiddotS

nii )T whereSi is the characteristic

size for the unit (area in m2 for heat exchangers) The cost factors (Cfixedi andCvariablei) and the scaling factorni are assumed constants for each unitT is thecapital depreciation time egT = 10years First if the structure of the network isfixed the size independent parameter does not matter (so we may useCfixedi = 0)Second we here consider only the compressor and heat exchanger costs (mainequipment) Third we assume that the exponentn = 1 for the compressors Wecan then add the operation and capital cost for the compressor into a singleterm

lowast∆Tmin in this paper refers to the individual exchanger approach temperature(ERAT) and notthe heat recovery approach temperature (HRAT) or ldquopinchrdquo temperature of the entire network


see Equation62 below Fourth we assume that the size dependent parameterand the exponent are equal for all heat exchanger (Cvariablei= C0 andni = n) Wechoose to fixn= 065 for the heat exchanger areas and useC0 as a single adjustableparameter

The resulting ldquosimplified TACrdquo design problem with the cost factorsC0 andk asparameters becomes

min

k middotWs+ pfeedmiddot mfeedminus pLNG middot mLNG minus pfuel middot mfuel︸︷︷︸

Jflows

+C0 middotsumi

Ani

[$year]

(62)

This minimization is subject to some design constraints see Section612 Ai isthe heat transfer area for the two hot streams in the main heat exchanger (naturalgas and warm refrigerant) and for the sea water heat exchanger Note thatk is notonly the price of the compressor energy as it also includes the annualized capitalcost for the compressor One may viewk[$MWyear] as the price of ldquorentingrdquo acompressor

With no capacity limits this problem (Equation62) is actually ill-posed as it isoptimal to have infinite feed (capacity) In practice one either designs fora givenfeed or imposes constraints that limit the capacity In this paper we assume thatwe wish to find the optimal design with a given maximum compressor shaft powerWmax

s The factork then does not matter as the termkWs is fixed We also donot include the condenser in the design and instead specify the outlet temperatureof the condenser A mass balance gives ˙mfuel = mfeedminus mLNG and we may writeJflows = mfeed(pfeedminus pfuel)minusmLNG (pLNG minus pfuel) =minusmLNG (pLNG minus pfuel) wherewe have assumedpfeed= pfuel This is reasonable since feed may be used as a fuelSinceWs is constrained the optimal design problem (Equation62) simplifies to

min(minusmLNG +C0

(A065

HOT +A065NG

))(63)

subject to Ws leWmaxs

mfuel le mmaxfuel

cle 0

The minimization is with respect to design parameters (AHOT andANG) operatingparameters (flows pressures splits etc) We may adjust the parameterC0 in Equa-tion 63to obtain a reasonable∆Tmin This will unlike the ldquo∆Tmin-methodrdquo (wherewe maximizemLNG with ∆Ti ge ∆Tmin as constraints) give the same operating pointif one re-optimizes the operation with the areas fixed at the optimal design valuesThe setc is constraints that may vary for the different cases

61 Introduction 85

612 Design constraints

In design it is necessary to impose some constraints for the optimization to assurethat a feasible solution is found

Pressure There are typically constraints on pressures in the process These con-straints may be related to equipment manufacturing limits For example alarge conventional centrifugal compressor has a maximum pressure in theorder of 30 to 40bar while a vertical split centrifugal compressor may haveoutlet pressure up to about 80bar (General Electric Oil and Gas January2007) Constraints may also arise from the fact that higher pressures are notdesirable from a safety point of view (eg explosion danger) One may alsoconstrain certain pressures to simplify the objective function by removingthe pressure dependence (higher pressure requires more expensive pipingheat exchangers etc)

Temperatures There may be constraints on temperature gradients (eg in heatexchangers) or on absolute temperatures Both of these may be related tomechanical issues such as stressThis has not been considered here

Compressor suction volume (Vsuc) In operationthe compressor suction volumeand compressor head are linked to the speed of rotation through the charac-teristic curve In designone may have to limit the compressor size to keepit within physical limits (eg by limiting the compressor suction volume)The current maximum limit for a single flow centrifugal compressor seemsto be 380000m3h-1 (General Electric Oil and Gas January 2007)

Compressor head A simple correlation for the maximum head (or specific en-thalpy rise) per compressor wheel is for a centrifugal compressor (see Equa-tion 173 on page 37 inLudtke 2004)

Head= ∆h = smiddotu2 [kJkg-1] (64)

wheresasymp 057minus066 is the work input factor andu[ms-1] is the velocity atthe wheel tip Adding the numbers for each wheel gives the total head forthe compressor For example one compressor wheel with 17m diameterand a rotational speedω = 3600RPM gives the following head

u =3600min-1

60smin-1middotπ middot17m= 320ms-1 (65)

∆h = (320ms-1)2 middotsasymp 58minus68kJkg-1 (66)

Table61gives the head for different rotational speeds and wheel diameters(s= 057) For large compressors the number of wheels is typically six orless (up to eight for the barrel type)


Table 61 Maximum head∆h[kJkg-1] per compressor wheel for different wheeldiameters and rotational speeds (Equation64 with s = 057) The last columnshows the total head for a compressor with one of each of the wheels in the table

Wheel diameterRotational speed 17m 16m 15m 14m 13m 12m Sum 6 wheels3600RPM 596 528 464 404 348 297 26363000RPM 414 366 322 280 242 206 1830

Compressor pressure ratio (Pr) The maximum compressor pressure ratio is analternative simple way of limiting the compressor head (egPrice andMortko (1996) report the value of 55 for their compressor) but this ap-proach is less exact since the head also depends on the mass flowrate Theuse of Equation64 is therefore preferred

Compressor shaft work (Wmaxs ) A maximum value for the compressor shaft work

may be imposed for example due to physical limitations in the driver to thecompressor or the compressor itself or in the available power supply

Mach number The gas velocity must be below the sonic velocity (Mach numberless than one) High Mach numbers are typically a problem at low pressuresso this is an issue at the compressor inlet However the velocity is onlyknown at the compressor outlet (the tip speed) so it is common to reportthe machine Mach number as the tip speed over the sonic velocity at theinlet The machine Mach number may be higher than one (as high as 125according toLudtke(2004)) without having an actual Mach number higherthan oneWe have not considered this constraint

LNG outlet conditions There may be constraints on the LNG heating value andcompositionThis is not considered here We have however assumed stor-age as saturated liquid atP = 11bar

Fuel specifications There may be constraints on the fuel heating value and com-positionThis is not considered here We have limited the maximum amountof fuel to a value somewhat larger than the energy needed by the refrigerantcompressor (by combustion in a gas turbine)

62 Process description

Figure61shows a simplified flowsheet of the PRICO processNatural gasis fedto the main heat exchanger (NG HX) after some pretreatment (removal of water

62 Process description 87

CO2 etc) which is not included in this paper The natural gas is cooled liquefiedand sub-cooled by heat exchange in NG HX with the cold refrigerant To furtherlower the temperature the sub-cooled liquid is then expanded and the liquid isseparated from the vapour (flash gas) and sent to storage as LNG The vapour(flash gas) is heated re-compressed and used as fuel but this part of the plant hasnot been included in this work Instead the flash gas is considered as a productwith a given fuel prize The further simplified process considered in this work isshown in Figure62

LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout

Figure 62 Further simplified flowsheet of the PRICO process used in this work

The refrigerant is partially condensed in the sea water (SW) cooler (condenser)The main part (all in our case see Figure62) of the refrigerant is fed to the NGHX and is cooled together with the natural gas stream The refrigerant is a sub-cooled liquid at the outlet of NG HX and is expanded to the low pressure (Pl ) Theresulting two-phase mixture provides the cooling in NG HX by vaporization Theoutlet from the heat exchanger (NG HX) is slightly super-heated partly toavoiddamage to the compressor We have assumed∆Tsup= 10C in most cases

We have made some simplifications compared to the process described inPrice andMortko (1996) First we do not consider removal of heavy components from thenatural gas feed This removal may be done upstream or integrated in the refriger-ation process (as shown byPrice and Mortko(1996)) A more detailed discussion


on this is given belowSecond we have not included the refrigerant separator andrefrigerant pump indicated byPrice and Mortko(1996) This however will notchange the thermodynamics if the pump work is neglectedThird as already men-tioned we have not considered the flash gas heating (Fuel HX) and compressionThe resulting flowsheet is shown in Figure62

Some of the data reported byPrice and Mortko(1996) which we have used asdesign constraints is shown in Table62 In addition we use the following

bull Composition of natural gas (mole-) 897 methane 55 ethane 18propane 01 n-butane and 28 nitrogen Note that this composition ismore methane rich than that reported inPrice and Mortko(1996) (897versus 832) The mole weight isMW = 00176kgmolminus1

bull The temperature after the refrigerant condenser and the temperature in thenatural gas feed are both 30C Price and Mortko(1996) report 29C and32C respectively

bull Pressure drops

ndash 5bar on natural gas side in main heat exchanger

ndash 01bar in SW cooler

ndash 4bar for hot refrigerant in main heat exchanger

ndash 1bar for cold refrigerant in main heat exchanger

ndash 03bar from main heat exchanger to compressor

bull Constant heat transfer coefficients

bull The refrigerant is a mix of nitrogen (N2) methane (C1) ethane (C2) propane(C3) and n-butane (nC4) and the composition is found by optimizationPriceand Mortko(1996) use hydrocarbons ranging fromC1 to iC5

bull The SRK equation of state is used for the thermodynamic calculations

63 Results for optimal design

The numerical results from the optimization for nine cases are reported in Table63 Boldface numbers indicate specifications or active contraints We have ad-justedC0 in Equation63to obtain∆Tmin asymp 20C in the main heat exchanger (NGHX) for all cases We have assumed 10C super-heating at the compressor inlet

63 Results for optimal design 89

Table 62 Design constraints based on data fromPrice and Mortko(1996)Pfeed[bar] 40Ws[MW] (max) 775Vsuc[m3s-1] (max) 88lowast

Ph [bar] (max) 22Pr [-] (max) 55

lowastThis was reported as 317m3s-1 but this is probably a misprint and should be 317000m3 h-1

(∆Tsup= 10C) except in Cases63and64 Note from the results that it is optimalin all cases to have noC3H8 in the refrigerant

Table 63 Optimal design results for nine different casesCase 61 62 63 64 65 66 67 68 69

mfeed[kgs-1] 522 450 453 448 496 496 514 761 808mLNG [kgs-1] 446 417 420 415 463 463 481 711 758mfuel [kgs-1] 77 333 333 333 333 333 333 50 50

mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120

Ph [bar] 220 220 220 220 504 300 370 300 300Pressure ratio[-] 55 55 55 55 225 166 117 73 72

Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161

Vsuc[m3s-1] 843 833 840 839 751 106 70 106 106UAHOT [MW C-1] 384 409 413 398 187 229 268 518 522UANG [MW C-1] 48 44 44 46 57 55 58 80 82UAtot [MW C-1] 431 453 457 444 244 284 326 598 604

C0 middot10minus3 [kgs-1m-13]lowast 110 120 130 107 37 51 54 3000 3000Refrigerant composition

xCH4 [mole-] 333 323 323 325 311 292 319 325 332xC2H6 [mole-] 353 332 334 347 323 329 327 329 335xC3H8 [mole-] 00 00 00 00 00 00 00 00 00

xnminusC4H10 [mole-] 250 246 243 228 267 303 252 234 235xN2 [mole-] 64 99 100 100 99 76 102 112 98

lowastC0 adjusted to obtain∆Tmin asymp 20C

Case 61Nominal design using data fromPrice and Mortko(1996) in Table62


We specify the LNG temperature at the exit of NG HX (Tout) to minus144C (Priceand Mortko 1996) The LNG production ( ˙m = 252kmoles-1 = 446kgs-1) isslightly larger (37) than that reported byPrice and Mortko(1996) (mLNG =243kmols-1lowast) but note that the feed composition is different and that we haveneglected the removal of heavy components On the other hand we have notincluded the heating of flash gas (in Fuel HX) before re-compression to turbinefuel which would have further increased the LNG production by providing somecooling for free

The resulting flash gas is 77kgs-1 and will produce about 230MW of energy bycombustion in a gas turbine (assuming 60 efficiency and 50MJkg-1) This is toohigh considering that the refrigerant compressor consumes less than 80MW In theremaining cases we have therefore limited the amount of flash gas to 333kgs-1 toachieve about 100MW equivalents of fuel which replaces the specification onTout

Case 62Constraint on the amount of flash gas after expansion (333kgs-1) suchthat it gives about100MWworth of energy in a gas turbine

Note that the temperature out of the heat exchanger (Tout) is reduced fromminus144to minus156C to reduce the amount of flash gas This results in a 60 reduction inproduction compared with Case61 (mLNG drops from 446kgs-1 to 417kgs-1)This is because we are unable to cool as much natural gas and this is not compen-sated for by the increased liquid fraction after expansion The effect of the outlettemperature (Tout) is further discussed below

Case 63Optimized super-heating

We find by removing the constraint on super-heating that∆Tsup increases from100C to the optimal value of 116C This gives an 08 increase in LNGproduction compared with Case62 This illustrates as discussed byJensen andSkogestad(2007) that the optimal super-heating is not zero for a system withinternal heat exchange

Case 64Higher degree of super-heating

In this case we specify a higher degree of super-heating (257C compared to theoptimal of 116C) This gives only a 13 reduction in LNG production com-pared to Case63 which shows that the optimum is ldquoflatrdquo in terms of super-heatingWith 022C super-heating we get a reduction of 23 in LNG production com-pared with Case63

lowastCalculated from 471MNm3day-1


In reality we expect that the heat transfer coefficient is lower in the super-heatingsection than in the vaporization section This suggests that the optimal degreeofsuper-heating will be lower than what we find with constant heat transfercoeffi-cients

Until now we have fixed the two refrigerant pressures Specifically thedischargepressurePh is fixed at 22bar and the pressure ratioPr = PhPl at 55 (Table63)However some authors have published optimization results with discharge pres-sure much higher than 220bar (Lee et al 2002 Del Nogal et al 2005) Theyalso claim that the refrigerant flowrate should be about 3-4 times the flowrate ofnatural gas on mole basis For the cases up till now we have obtained a ratio ofabout 6 which is about 50 higher These two observations are closelyrelated asthe amount of refrigerant depends on the pressure ratio (Pr)

Case 65No pressure constraints

Here we optimize the process without the constraint on discharge pressure andpressure ratio We see that the production is increased from 417kgs-1 to 463kgs-1

(11) while the refrigerant amount is reduced from 147kmoles-1 to 764kmoles-1

(which gives a ratio 29 between refrigerant and LNG flowrate) To achieve thisthe high pressure is increased toPh = 504bar and the pressure ratio isPr = 22

Some other interesting results to note are

bull the compressor suction volume decreases

bull the necessary heat transfer area for the warm refrigerant stream isless thanhalfUA is 188MWC compared to 409MWC for Case62

Both these effects are related to the fact that much less refrigerant is needed buthow can this be explained The cooling duty per kg of refrigerant is closely re-lated to the compressor head[kJkg-1] which again is closely related to the pres-sure ratio So increasing the compressor head (and pressure ratio) willincreasethe cooling duty per kg of refrigerant and thus decrease the required amount ofrefrigerant

There is a potential problem with this design A high pressure ratio usually requiresmore compressor stages (casings) and this may not be desirable althoughsome ofthe extra capital cost related to the extra compressor casing and higher pressurewill be offset by the reduction in heat transfer area

We wish to limit the PRICO process to one compressor casing which may not befeasible with the high pressure ratio of 22 in Case65 To get a realistic design weuse performance specifications for the MCL1800 series compressor from GeneralElectric Oil and Gas(January 2007)


Case 66MCL1800 series compressor

MCL1800 is a centrifugal compressor with casing diameter of 1800mm The re-ported maximum suction volume is 380000m3h-1 or about 106m3s-1 the maxi-mum discharge pressure is 30bar and the maximum shaft work is 120MW (Gen-eral Electric Oil and Gas January 2007) In this case we keep 775MW as themaximum compressor shaft work to compare with the other cases and specifya maximum compressor suction volume of 106m3s-1 and maximum pressure of30bar

Interestingly the results show that we are able to almost match the production andpressure ratios obtained in Case65with realistic specifications and one compres-sor casing The total head in the compressor may be achieved with one compressorcasing with 5 wheels and a rotational speed of 3600RPM see Table61

Note that the suction volumeVsuc is an active constraint for the three last cases inTable63where actual compressor data are utilized

The MCL1800 compressor seems a bit large for the specified duty so let us try asmaller compressor


MCL1400 is a centrifugal compressor with a casing diameter of 1400mm Thereported maximum suction volume is 250000m3h-1 or about 70m3s-1 the maxi-mum discharge pressure is 37bar and the maximum shaft work is 75MW (GeneralElectric Oil and Gas January 2007) We have assumed that it is possible to use775MW power

We get a slightly higher production (36) compared with Case66 This illus-trates that the increase in outlet pressure (from 30bar to 37bar) more than compen-sates for the reduction in compressor suction volume (from 106m3s-1 to 70m3s-1)The reqired head is slightly reduced because of a higher suction pressure How-ever the maximum head is reduced compared with the MCL1800 compressorbecause of a reduced tip speed The total achievable head in the compressor isstrongly affected by the reduction in wheel diameter throughout the compressorcasing Using a 70cm reduction from one wheel to the next and an intial wheeldiameter of 135m gives an estimated maximum total head of 172kJkg-1 with 6compressor wheels This is significantly less than the required amount and we be-lieve that the design in Case67 is infeasible with one compressor casing but adetailed compressor design is necessary to verify this conclusion

Finally we would like to find the maximum train capacity limit for the PRICOprocess with a single compressor casing


Case 68Again we utilize the larger MCL1800 but we allow for more shaft powernamely120MW

We find that we may produce 711kgs-1 LNG in a single PRICO train with onecompressor casing using realistic design data Note that the required compressorhead is reduced from 216kJkg-1 to 162kJkg-1 compared to Case66 so for thiscase we may use a slower driver (see Table61) for example a Frame 9 gas turbinewith rotational speed of 3000RPM (General Electric Oil and Gas January 2007)

Note that the cost factorC0 = 3000 is increased tenfolds compared to the othercases This was necessary to achieve∆Tmin = 20C There seems to be no prac-tical reasons for this large increase in the cost factor and an alternativedesignoptimization is given below

1copy 2copy

Figure 63 A flowsheet of the PRICO process illustrating the use of liquid turbineand valve as expansion device

The LNG expansion valve and the refrigerant expansion valve shown inFigure61may be exchanged with a combination of liquid turbine and valve see Figure63Ideally one would do the entire expansion in a turbine but two-phase turbines areto our knowledge not in use so it is necessary to use the combination of a liquidturbine and a valve (Barclay and Yang 2006) The liquid turbine will then take thepressure down to slightly above the saturation pressure and the expansion valvewill take care of the two-phase expansion The main advantage with this solutionis not that power is generated by the liquid turbine (this is actually quite small)but rather the extra cooling provided by the isentropic expansion in the turbinecompared to isenthalpic expansion in the valve This reduces i) the amount of


flash gas that does not contribute to the product for LNG expansion andii) theamount of flash gas that does not contribute significantly to the cooling butstillneeds to be compressed by the compressor for the refrigerant expansion

Case 69A liquid turbine is included in the expansion of the natural gas1copy andin the expansion of the refrigerant2copy

The production is further increased by 66 compared with Case68 The totalheat transfer area is increased by 10 Note that also for this case we had tospecify a highC0 to obtain∆Tmin asymp 20C

We have assumed an isentropic efficiency of 80 in the liquid turbines We haveused a back-off from saturation at the turbine outlet of 20bar A turbine will beeven more advantageous for cases with higher pressures both in the refrigerantcycle and in the natural gas stream

In summarywe see that some design constraints strongly affect the optimal solu-tion These constraints are related to the compressor performance maximumsuc-tion volume maximum discharge pressure maximum head and maximum shaftwork The changes given by these constraints are illustrated in cases65 to 68Other constraints have less influence on the optimal solution these are the degreeof super-heating and the amount of flash gas (determined by the temperature aftercooling or the amount of flash gas) The changes given by these constraints areillustrated by the first four cases

631 An alternative design optimization

Above we adjusted the cost factorC0 to achieve∆Tmin asymp 20C For Case68 thisresulted in a cost factor unrealistically much larger than for the other cases Wesuspect that this is due to the non-linear behaviour of∆Tmin A better approachmay be to fixC0 HereC0 = 110middot103kgs-1

(m-2)065

is used for all cases

Table 64 shows key results for all nine cases with the same specifications forsuper-heating maximum work and pressures as before Note that for Case68weget an increase by 86 in the LNG production compared to the correspondingcase in Table63 whereC0 = 3000middot 103kgs-1

(m-2)065

This is achieved by in-creasing the total heat transfer area by 247 and∆Tmin is reduced from 20C to15C A similar increased production and heat transfer areas is achieved for Case69 For cases65to 67the production is reduced compared to the results in Table63

The results in Table64 are obtained more easily than the results in Table63

64 Discussion 95

Table 64 Optimal design withC0 = 110middot103kgs-1(m-2)065

Alternative Case 61 62 63 64 65 66 67 68 69mLNG [kgs-1] 446 454 458 443 365 426 451 772 814

∆Tmin [C] 196 196 186 204 289 251 237 148 165UAHOT [MW C-1] 384 417 423 397 80 184 225 642 628UANG [MW C-1] 48 46 47 45 29 41 46 103 104UAtot [MW C-1] 431 464 469 442 109 225 271 746 732

Change compared with Table63∆mLNG [] 00 89 90 -11 -212 -80 -62 86 74∆UAtot [] 00 24 26 -045 -555 -210 -169 247 212

where it was necessary to adjustC0 until we got∆Tmin asymp 20C

In terms of optimality it seems better to fix the cost factorC0 rather than the∆Tmin

In Cases62 and63 the relative (percent) increase in production ( ˙mLNG) is morethan three times the relative increase in heat transfer area (UAtot) even though thework Ws is constant This indicates strongly that the cost factorC0 is too high Analternative approach would therefore be to adjustC0 such that the relative increasein mLNG andUAtot are similar

64 Discussion

641 Compressor

The number of impellers or wheels in the compressor will affect the achievablehead Ludtke (2004) reports that centrifugal compressor may be manufacturedwith up to ten impellers in one casing but that more than five impellers usuallygives a cost in terms of reduced efficiencyGeneral Electric Oil and Gas(January2007) indicates that their compressors may be delivered with up to eight impellersBased on the numbers for maximum head per impeller in Table61 we have nodifficulty achieving the necessary head for the cases presented with the MCL1800compressor see Table63 However there is some uncertainty

bull The wheel diameter is decreasing through the compressor and finding theoptimal design requires a detailed analysis of the compressor

bull The number of wheels may affect the efficiency which we have assumedconstant at 828


Price and Mortko(1996) report that the PRICO process uses an axial compressorThe efficiency may be slightly higher for an axial compressor than for a centrifugalcompressor

642 Heavy extraction

We have neglected the process for removing heavy components from the naturalgas stream The paper byPrice and Mortko(1996) shows that this process is in-tegrated in the refrigeration process as shown in Figure64(a) The natural gas istaken out of the heat exchanger after being cooled somewhat The heavy compo-nents are then in the liquid phase and is extracted (methane and ethane in the liquidis recovered and mixed with the LNG product) The vapour stream is sent back tothe heat exchanger for further cooling

Alternatively one may have the removal of heavy components before the refriger-ation process This is shown in Figure64(b)

The choice between upstream and integrated NGL recovery depends onseveralfactors and will not be treated here

The fraction of NGL in the natural gas feed is quite small for the case we haveconsidered so we believe inclusion of an integrated NGL extraction will nothavea large impact on our results

(a) Integrated NGL recovery (b) Upstream NGL recovery

Figure 64 Alternative locations of the NGL recovery

64 Discussion 97

643 Feed pressure

Higher feed pressure gives higher maximum production because the main part ofthe cooling will be at a higher temperature (since the condensation temperaturerange is higher with increased pressure) Figure65shows the maximum produc-tion as function of feed pressure However this effect is not for free since a feedcompressor is needed For example compressing the feed from 40bar to60barrequires a compressor with about 53MW power for the natural gas flowrate inCase68 To find the optimal feed pressure it is necessary to also consider thefeed compressor and the extra capital cost related to higher pressure inthe heatexchanger and piping

The feed pressure increase is limited if the NGL extraction is integrated (becausethe separation in NGL extraction is harder at higher pressure eg propane has acritical pressure of about 42bar) so very high feed pressures areonly feasible forplants with NGL extraction prior to refrigeration

P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100

Figure 65 Maximum LNG production as function of feed pressure for Case69

644 Large PRICO plants

Larger production rates requires more than one compressor so one then has thechoice of using more PRICO trains in parallel or to use more cycles in cascade withone compressor for each cycle An obvious advantage with choosing more parallelPRICO trains is that one may operate on part load if one compressor shuts downBut important issues are thermodynamic efficiency and capital cost compared withcascaded cycles

Based on Case69 in Table 63 we get a maximum LNG production of about


76kgs-1 see Figure65 with P = 40bar This is about 22MTPA based on 335operating days per year so two trains in parallel gives about 44MTPA while threetrains in parallel gives about 66MTPA However if a large plant is constructedone may wish to have the NGL extraction up front and install a feed compressorprior to the refrigeration process From Figure65 we see that a feed pressure of60bar gives a maximum production of about 97kgs-1 or 28MTPA per train Thisresults in a maximum production of 56MTPA and 84MTPA for two and threeparallel trains respectively Even higher feed pressures may be feasible

The specific work is 439kWht-1 for 40bar feed pressure and 344kwht-1 for 60barfeed pressure for the process with liquid turbine for expansion of LNG and refrig-erant This is without considering the shaft work generated by the turbines andrequired by the feed and fuel compressor

Roberts et al(2004) presents different driver configurations for the AP-X processfrom the Air Products company The configuration with the highest production(10MTPA) is with three Frame 9 gas turbines and and additional 10MW fromthe helperstarter motor giving a total of 400MW We have assumed 130MWfor each Frame 9 gas turbine this the ISO rating (General Electric Oil and GasJanuary 2007) With the same available shaft work we would achieve

28MTPA120MW

middot400MW= 93MTPA

for the PRICO process with 60bar feed pressure and liquid turbine for expansionof both LNG and refrigerant This is 7 lower for the same shaft work but theactual numbers will depend heavily on several factors such as feed pressure feedtemperature feed composition and actual compressor performance

65 Conclusion

An objective function for design optimization has been derived for a PRICO pro-cess The process is optimized for several different constraints and compared withthe commercial process and other publications Using compressor specificationsfound online we are able to increase the LNG production compared to the com-mercial process Important constraints especially concerning the compressor fea-sibility are discussed Finally we found that the PRICO process has about 7less production than the AP-X process with the same available shaft power

Bibliography 99

Bibliography


Del Nogal F Kim J Smith R and Perry S J (2005) lsquoImproved design ofmixed refrigerant cycles using mathematical programmingrsquoGas Processors As-sociation (GPA) Europe Meeting Amsterdam

General Electric Oil and Gas (January 2007) lsquoLiquefied natural gas En-hanced solutions for LNG plantsrsquowwwgepowercombusinessesge_oilandgasendownloadsliquified_natural_gaspdf

Jensen J B and Skogestad S (2007) lsquoOptimal operation of simple refrigera-tion cycles Part I Degrees of freedom and optimality of sub-coolingrsquoComputChem Eng31 712ndash721

Lee G C Smith R and Zhu X X (2002) lsquoOptimal synthesis of mixed-refrigerant systems for low-temperature processesrsquoInd Eng Chem Res41(20) 5016ndash5028

Ludtke K H (2004)Process centrifugal compressors Springer-Verlag BerlinHeidelberg


Roberts M J Liu Y Bronfenbrenner J C and Petrowski J M (2004) Reduc-ing LNG capital cost in todayrsquos competitive environmentin lsquoLNG 14 confer-ence Doha Quatarrsquo

Stebbing R and OrsquoBrien J (1975) An updated report on the PRICO(TM) pro-cess for LNG plantsin lsquoGASTECH LNG Natural Gas LPG international con-ference Parisrsquo

100 Bibliography

Chapter 7

Optimal operation of a simpleLNG process

Considering the large amount of work that goes into the design of LNGprocesses there is surprisingly little attention to theirsubsequent oper-ation This partly comes from the assumption that optimal design andoptimal operation are the same but this is not generally true In this pa-per we study the optimal operation of a relatively simple LNGprocessnamely the PRICO process We find that the process has four opera-tional degrees of freedom (neglecting the degrees of freedom related torefrigerant composition) We then study the two modes of operation i)given production and ii) maximum production

71 Introduction

The process considered in this paper is a single mixed refrigerant process namelythe PRICO process (Stebbing and OrsquoBrien 1975 Price and Mortko 1996) Thisis the simplest configuration used commercially for liquefaction of natural gasandit has been optimized in several publications (Lee et al 2002 Del Nogal et al2005 Chapter6) but only with respect to designSingh and Hovd(2006) studythe controllability of the process but they do not consider optimal operation whichis the theme in this paper

An important issue in plantwide control is to find the degrees of freedom thatmaybe used for online optimization (Skogestad 2002) In our case these are the sameas the steady-state operational degrees of freedom and this number is important forseveral reasons First it determines the degrees of freedom availablefor solving

101

102 Optimal operation of a simple LNG process

the optimization problem However more importantly in terms of operation it de-termines the number of steady-state controlled variables that need to be selectedOptimal operation is normally implemented by keeping the selected variables atconstant setpoints Note that the selection of controlled variables is equally im-portant if we use a model-based control structure such as model predictive control(MPC)

There are two main modes of operation for a LNG process

1 With a given LNG production (load) minimize the compressor shaft workWs (optimize efficiency)

2 Maximize the LNG production rate subject to given constraint (maximumshaft workWmax

s )

In general to implement optimal operation we first need to control active con-straints Second we need to find controlled variables for the unconstrained degreesof freedom We here use the self-optimizing control approach ldquoSelf-optimizingcontrol is when we can achieve acceptable loss with constant setpoint values forthe controlled variables (without the need to re-optimize when disturbances oc-cur)rdquo (Skogestad 2000)


Figure71shows a simplified flowsheet of the PRICO process The PRICO processworks as follows After compression to pressurePh the mixed refrigerant is cooledto 30C in a sea water (SW) cooler before it is further cooled together with thenatural gas in the main heat exchanger The high pressure sub-cooledliquid is thensent through a liquid turbine and a choke valve to give a low-temperature saturatedliquid at pressurePm in the receiver The liquid is further expanded to low pressurePl to give a two-phase mixture which is vaporized in the main heat exchanger toprovide the necessary cooling duty The vapour is slightly super-heatedbeforeit is compressed back to the high pressure The PRICO process is discussed inmore detail in Chapter6 We here consider Case69 from Chapter6 where aliquid turbine is included both in the expansion of natural gas and expansionof therefrigerant

Note that the refrigerant is only partially condensed at pressurePh in the sea water(SW) cooler so both liquid and vapour are fed to the main heat exchangerWe haveplaced the liquid receiver at an intermediate pressure (Pm) before the choke valve


The extra choke valve between the liquid turbine and receiver is to give a safetymargin to saturation at the liquid turbine outlet

NG30C40bar

Ph

Pl

Pm

N

minus157C35bar Choke valveTurbine

Main heat exchanger


SW

30C

Figure 71 A simplified flowsheet of the PRICO process

Note that we have not included any extraction of heavy components from the nat-ural gas feed

721 Nominal conditions

bull The natural gas enters after pretreatment with a pressure of 40bar and atemperature of 30C

bull Composition of natural gas (mole-) 897 methane 55 ethane 18propane 01 n-butane and 28 nitrogen

bull Pressure drops

ndash 5bar in natural gas stream





ndash 03bar for the compressor suction

bull Constant heat transfer coefficients (UA)

bull The refrigerant is a mix of nitrogen (N2) methane (C1) ethane (C2) propane(C3) and n-butane (nC4) and the composition is used as a degree of freedomin the optimization

bull Cooling of refrigerant to 30C in SW cooler

bull Maximum shaft work in the compressorWmaxs = 120MW

bull We have not considered the turbine characteristic and we have assumed aconstant pressure drop of 20bar across the valve after the turbine Thisvalve is necessary to guarantee that no vapour is formed in the liquid turbine

722 Manipulated inputs

There are 9 manipulated inputs with a steady-state effect (potential controldegreesof freedom for controlu0)

1 Compressor rotational speedN

2 Choke valve valve openingz

3 Turbine rotational speed

4 Cooler flow of sea water (SW) in cooler

5 Load Feed flow of natural gas (can also be considered a disturbance)

6-9 Composition of refrigerant (5 components give 4 independent compositions)These degrees of freedom are not considered for control but need to be op-timized at the design stage

Assuming maximum cooling of refrigerant in the SW cooler or rather fixingT =30C after the SW cooling which consumes 1 degree of freedom this leaves 8degrees of freedom To find the nominal optimal steady-state operating point wewill use all 8 degrees of freedom but during operation we will assume constantrefrigerant composition so there are only 4 steady-state control degrees of freedom


723 Operational constraints

In general there are many constraints (represented by the equationc le 0 in theoptimization problems) that must be satisfied during operation

bull Super-heating (∆Tsup) The stream entering the compressor must not containliquid (but note that this is not necessary an active constraint in this casebecause there is internal heat exchange so it is actually optimal with somesuper-heating see Chapter6)

bull ToutLNG Natural gas temperature out of the main heat exchanger should be

within certain bounds This temperature sets the amount of flash gas andaffects the composition of flash gas and LNG

bull Refrigerant pressure Maximum bound (not considered in this paper)

bull Compressor outlet temperature Maximum bound (not considered in thispaper)

bull Compressor power (Ws) We assume maximum at 120MW see Chapter6

bull Compressor rotational speed (N) We assume maximum at 100

bull Compressor surge The compressor may in theory be operated in the surgeregion using active surge control (Gravdahl and Egeland 1999) but nor-mally one would like to operate with a certain margin to surge In this paperwe use the peak of the compressor characteristic curve as the limit (see Fig-ure 73) We define the variable∆msurge to be the distance from the peakof the characteristic curve and use the value∆msurgege 00kgs-1 (ie noback-off)

bull All flows must be non-negative and also have upper bounds

In particular the cooling water flow has a maximum value and it is clear fromphysical insight that maximum cooling is optimal (active constraint) Assumingthat we have a large area in this heat exchanger we will in the following replacethis constraint by the following

bull Maximum cooling Assume refrigerant hasT = 30C after SW cooler

With the assumption ofT = 30C after the SW cooler (flow of sea water at maxi-mum) we are left with 8 steady-state degrees of freedom (4 for control)


73 Model

When switching from design simulations to operation it is necessary to reformu-late parts of the models because the equipment is fixed Our goal is to have simplemodels that capture the most important operational effects We will here discussthe features that we have included in the operational models and also brieflymen-tion the effects that we have not considered

The process is modelled using the gPROMS software with the accompanying Mul-tiflash package for thermodynamic calculations The SRK equation of state isused for thermodynamic calculations both for the natural gas and the refrigerantThe main heat exchanger is a distributed system which for modelling purposeshas been discretized into 100 cells using forward and backward finite differencemethod for the cold and hot streams respectively

The pressure drop at the compressor inlet (suction) is modelled as

∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)

The remaining pressure drops are assumed constant The structure of the modelequations are summarized in Table73

731 Compressor characteristic

In operation one normally uses compressor characteristic curves to model thecompressor behaviour These curves relating flow efficiency pressure increaseand rotational speed are normally supplied by the compressor vendor ofthe in-stalled compressor but since we do not have a specific design and vendor we needa more general approach

Non-dimensional groups

Compressor characteristics are easiest represented using non-dimensional groupsFor exampleSaravanamuttoo et al(2001) assume the following functional depen-dencelowast

f (DNmP1P2T1T2 R) = 0 (72)

lowastThey actually use the two groupsRT1 andRT2 instead of our three termsT1 T2 andR

73 Model 107

where the characteristic lengthD [m] is usually taken as the compressor wheeldiameterN [s-1] is the rotational speed ˙m[kgs-1] is the mass flowrateP[kgms-2]is the pressureR = RMW [Jkg-1K-1] is the specific gas constant andT [K] isthe temperature Subscripts 1 and 2 represent the compressor inlet and outletrespectively Using the Pi theorem of dimensional analysisSaravanamuttoo et al(2001) reduce these 7 function dependencies to 4 independent non-dimensionalgroups

Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)

(pressure ratio temperature ratio reduced flow and reduced speed) From thesefour groups it is possible to express one group in terms of the remaining threebut Saravanamuttoo et al(2001) claims that the groupsP2

P1and T2

T1may be plotted

against the last two groups The justification for this (which we could not find inSaravanamuttoo et al(2001)) is probably that the outlet conditions of the compres-sor (T2 andP2) should depend on the inlet conditions only which are expressed bymr andNr

It is common to report the isentropic efficiencyη instead of the temperature riseT2T1

so the following dependencies are used to quantify the steady-state operation ofthe compressor

Pr = f (mrNr) (74)

η = f (mrNr) (75)

Characteristic compressor curves

The dependencies in74 and75 are normally given graphically as ldquocurvesrdquo butwe are here looking for simple algebraic relationships

We use the method ofMoore and Greitzer(1986) cited inGravdahl and Egeland(1999) with some adjustmentsMoore and Greitzer(1986) proposed to use a cubicequation to predict the characteristic curve of a compressor The equation for thepressure ratio is (using our own nomenclature)

Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1

is the pressure ratio over the compressor (the first non-dimensionalgroup in Equation73) Pr0 is the shut-off value for the compressor (the pressure


ratio delivered at zero flow)H andW are called the semi-height and semi-widthof the characteristic curve respectively and ˙mr is the reduced mass flowrate (thesecond non-dimensional group in Equation73)

The cubic equation has three parameters (Pr0 H andW) which physical signifi-cance are indicated in Figure72 The curve is for a given reduced rotational speedNr (the third non-dimensional group in Equation73)

2H

2W

Pr0

Pr [-]

mr[-]

Figure 72 A cubic compressor characteristic curve for a constant reduced com-pressor speed

The surge point where dynamic instability occurs is somewhere near the peak valueof the pressure ratio in Figure72 Operation to the left of this point is unstablewithout active (feedback) control This is discussed extensively byGravdahl andEgeland(1999) Note that the surge point is normally close to the ldquooptimalrdquo oper-ating point with peak pressure ratio and peak efficiency

To get the entire compressor map for all values of the reduced speedNr we pro-pose the following dependency onNr for the parameterH andW

H = H0minus12

(

H0 +Pr0

2minus1

)

middot (1minusNr) (77)

W = W0(Nr)13 (78)

This is by no means an exact approach but we get compressor characteristic curvesthat are similar to typical example curves shown in textbooks (egSaravanamuttooet al 2001)

For the isentropic efficiency we propose to use the following function

η = η0

((

1minus

(H minusH0

H0

)2)

minus1000(mrminus2W)2

)

(79)

73 Model 109

This captures the two most important effects the efficiency has a peak value fora given reduced rotational speed and the peak value is slightly affected by therotational speed

A sample compressor map is shown in Figure73 To compute ˙mr we have useda compressor with wheel diameterD = 17m and a working fluid with molecu-lar weightMW = 0032kgmol-1 and the following values for the parameters inEquation76to Equation79

bull Pr0 =minus29 Note that we have used a negative value for the pressure ratio atzero flowrateGravdahl and Egeland(1999) state thatPr0 gt 0 but we areonly interested in the operating regime to the right of the peak value so thisparameter does not have any physical meaning in our model We have useda high negative value to get a steeper characteristic curve

bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-

Figure 73 Compressor map for the refrigerant compressor using Equations76-79with N in the range 10 to 100 and nominal inlet temperature The red dotsindicates the peak for the pressure ratio curve


74 Objective function

The objective function for optimal operation is simpler than for optimal designdiscussed in Chapter6 because the investments are already made and the capi-tal cost does not need to be considered The cost function to be minimized thenbecomes

Joperation=pWs middotWsminus pWsturbine middotWsturbine+ pSWmiddotQC (710)

minus pLNG middot mLNG + pfeedmiddot mfeedminus pfuel middot mfuel

We make the following assumptions

bull Same price for fuel and feed Thenpfeedmiddotmfeedminuspfuel middotmfuelminuspLNG middotmLNG =(pfeedminus pLNG) middot mLNG = pLNG middot mLNG

bull Neglect income from turbine workpWsturbine = 0

bull Neglect cost of coolingpSW = 0

The optimization problem then becomes

minu

Wsminus pLNG middot mLNG (711)

subject to cle 0

Herecle 0 represent the mathematical formulation of the operational constraintsand the model equations

Depending on product prize and other external factors there are two main operatingmodes

Mode I Given throughput With a given feed flowrate or given LNG productionthe optimization problem simplifies to

minu

Ws (712)

subject to mfeed= given (or mLNG = given)

cle 0

Mode I will result in the same optimal operation as the nominal optimal de-sign provided the optimal design is done correctly (eg using the simplifiedTAC method in Chapter6) andmfeed is kept at the nominal feedrate

Mode II Maximum throughput If the LNG prize ( ˆpLNG) is sufficiently high andthere is no active constraint related to available feed or product distribution

75 Nominal optimum given production case (Mode I) 111

Table 71 The nominal operating point for Mode I - Given production Mode II -Maximum production

Mode I Mode IIWs[kW] Compressor work 106 120Ph [bar] Cycle high pressure 268 300Pl [bar] Cycle low pressure 367 414N [] Compressor rotational speed 100 100η [] Compressor efficiency 828 828

mLNG [kgs-1] LNG flowrate 698 767mREF[kgs-1] Refrigerant flowrate 549 614

∆msurge[kgs-1] Surge margin 0000 0000∆Tsup[

C] Super-heating before compressor 129 113Tout[

C] NG temperature after cooling -157 -157xCH4 [mole-] Methane in refrigerant 319 327

xC2H6 [mole-] Ethane in refrigerant 352 343xC3H8 [mole-] Propane in refrigerant 00 00

xnminusC4H10 [mole-] nButane in refrigerant 247 233xN2 [mole-] Nitrogen in refrigerant 82 97

Boldface Specifications and active constraints

(sommaxLNG is not an active constraint) then it will be optimal to maximize the

production of LNG and the objective function may be simplified

minu

minus mLNG (713)

subject to cle 0

Note that the operation in this mode may be quite different from the ldquonomi-nalrdquo optimum found for mode I

75 Nominal optimum given production case (Mode I)

Since the production rate (or feed rate) is fixed there are 7 steady-state degrees offreedom including the 4 refrigerant compositions

With a given production rate ˙mLNG = 698kgs-1 the nominal optimum is found bysolving the optimization problem in Equation712 The results are summarized inthe left column of Table71 The work in712was minimized with respect to the7 degrees of freedom including the 4 refrigerant compositions Note thatwe have


assumed that the refrigerant is 30C after the SW cooling The optimal operationof the compressor was found subject to compressor maps (Figure73) We findthat the following constraints are active at the nominal optimum

1 Temperature of natural gas after cooling at maximum (Tout = minus157C)

2 Surge margin at minimum (∆msurge= 0)

3 Compressor speed at maximum (N = 100)

Thus at the nominal optimum the only unconstrained degrees of freedom are therefrigerant compositions

76 Nominal optimum maximum production case (ModeII)

Here we consider mode II where maximum production is the objective see Equa-tion 713 Since the production rate (or feed rate) is free there are 8 steady-statedegrees of freedom (with 30C after SW cooling) For numerical reasons opti-mal operation in mode II was found by solving mode I for increasing values ofmLNG = given until no feasible solution was found

h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107

Figure 74 Optimal (nominal) pressure enthalpy diagram for the maximum pro-duction case (mode II)

76 Nominal optimum maximum production case (Mode II) 113

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

Figure 75 The circle shows the optimal (nominal) compressor operating point forthe maximum production case (mode II)

The nominal optimum including the optimal composition of the refrigerant issummarized in the right column of Table71 The cycle is illustrated in a pressureenthalpy diagram in Figure74 and the optimal temperature and temperature dif-ference profiles in the main heat exchanger are shown in Figure76(a)and76(b)respectively

We find that the following constraints are active at the nominal optimum

1 Compressor work at maximum (Ws = 120MW)



4 Compressor rotational speed at maximum (N = 100)

Note that there are two ldquocapacityrdquo constraints that are active (1 and 4) Again theonly unconstrained degrees of freedom are related to the refrigerantcomposition

We have now identified the nominal optimum for the two cases but how should


Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50

(a) Temperature profile

Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30

(b) Temperature difference profile

Figure 76 Optimal temperatures in the heat exchanger for the maximum produc-tion case (mode II)

we control the process to maintain close to optimal operation when the processisexposed to disturbances This is discussed next

77 Optimum with disturbances

Table 72 Nominal minimum and maximum values for the disturbances Thenumbers in parentheses are for mode I

Nominal Min Max NameWmax

s [MW] 120 110 130 d1lowast

Pfeed[bar] 40 35 45 d2

Tin [C]dagger 30 25 35 d3

xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7

mLNG [kgs-1] 698 665 731 d8Dagger

lowastOnly used for mode IIdaggerThe temperature of natural gas and refrigerant at the inlet to the main heat exchangerDaggerOnly used for mode I

The next step is to consider optimal operation with disturbances and we considerthe eight disturbance variables given in Table72 We here fix the refrigerant com-

77 Optimum with disturbances 115

position because it is assumed to be unavailable as a degree of freedom duringnormal operation (instead the composition is treated as a disturbance see Table72) With a fixed temperature (30C) after the SW cooler there are then 4 remain-ing degrees of freedom During operation it is always optimal to cool the naturalgas tominus157C (to avoid product give-away) and one degree of freedom is spentto set the load

Mode I The production rate is given

Mode II In the maximum production case it is always optimal (for all distur-bances) to operate the compressor at its maximum (Ws = Wmax

s )

Thus two constraints are always active and this leaves for both modes only 2steady-state operational degrees of freedom and the resulting optimal operationconditions for various disturbances are summarized in Table75 The results arealso shown graphically as dots in Figure77 and Figure78 for mode I and modeII respectively

Recall that the surge margin constraint (∆msurge= 0) and compressor maximumspeed constraint (N = Nmax) were active in the nominal point and we find as onewould expect that these remain active for most of the disturbances but not allAlso note that some disturbances are not feasible in mode I probably because ofthe fixed refrigerant composition

To obtain optimal operation we should always implement the active constraintsand then find rdquoself-optimizingrdquo variables for the remaining unconstrained degreesof freedom in each region Strictly speaking to be truly optimal we then needtoconsider four regions

1 N = Nmax and∆msurge= 0 is optimal (two active constraints implementationis obvious)

2 ∆msurge= 0 andN lt Nmax is optimal (unconstrained optimum ie need tofind an associated controlled variable forN)

3 N = Nmax and∆msurgegt 0 is optimal (unconstrained optimum ie need tofind an associated controlled variable for∆msurge)

4 N lt Nmax and∆msurgegt 0 is optimal (unconstrained optimum ie need tofind an associated controlled variable forN and∆msurge)

All this cases can occur as seen in Table75 This becomes rather complicatedFirst a large effort is required to find the best self-optimizing variables in each ofthe three last regions Second even if we can find the self-optimizing variables ineach region it is not clear when to switch between the regions (that it it is easy to


identify when to switch when we encounter a constraint eg going from region 2to 1 but it is more difficult to determine when to leave a constraint region)

For practical implementation we would therefore prefer to have the same con-trolled variables in all regions and in our case the obvious policy is to considerkeeping the variables at constraint (∆msurge= 0 andN = 100) in all regions Ob-viously this is not optimal but the loss is rather small as discussed next except forsome cases in mode I where it seems operation is not feasible (without changingthe composition or feed rate)


A preliminary screening was performed by using the maximum scaled gain method(Halvorsen et al 2003) Some variables where discarded based on these results(eg the degree of sub-cooling cycle high pressure) Also we found that thesurge margin (∆msurge) is a much more promising controlled variable than any ofthe alternatives we tested Thus we choose to fix∆msurge= 0 and this gives onlyminor losses as seen below

Figure77 shows compressor shaft work as a function of 6 of the 7 disturbances(d3 minus d8) considered for mode I∆msurge= 0 and the following controlled vari-ables are tested for the remaining degree of freedomN = 100∆Tsup= 129CPl = 367barTout

com = 126C andmref = 549kgs-1 The dots shows re-optimizedoperation where bothN and∆msurgehas been optimized Note that the composi-tion of the refrigerant is still the same as for the nominal operating point The plotsare obtained by ldquobruteforce evaluationrdquo which involves fixing the variables andcomputing the resulting operating point for varying disturbances The disturbanceis plotted on the X-axis with the nominal value at the center and the Y-axis showsthe corresponding compressor shaft work in MW

For the four last disturbances (d5minusd8) we are not able to find a feasible solutionfor some values of the disturbances This means that we are not able to satisfythe specified production rate with the given refrigerant composition The best con-trolled variable to maintain constant is the compressor rotational speedN Forsome disturbances is it the only feasible controlled variable and it remains close tooptimal for all operating points

Figure78shows LNG production as a function of 6 of the 7 disturbances (d1d2d4minusd6) considered for mode II∆msurge= 0 and the following alternative controlledvariables are tested for the remaining degree of freedomN = 100 ∆Tsup =113C Pl = 414barTout

com = 124C andmref = 614kgs-1 The dots shows re-


TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N

(a) Disturbanced3 = Tin

XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN

(b) Disturbanced4 = xCH4

XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom

(c) Disturbanced5 = xC2H6

XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup

(d) Disturbanced6 = xC4H10

XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom

(e) Disturbanced7 = xN2

mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom

(f) Disturbanced8 = mLNG

Figure 77 Shaft work for different control structures as function of disturbancesfor mode I Dots are re-optimized operation with constant composition


d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN

(a) Disturbanced1 = Wsmax

d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f

(b) Disturbanced3 = Tin

d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref

(c) Disturbanced4 = XCH4

d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref

(d) Disturbanced5 = XC2H6

XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup

(e) Disturbanced6 = XC4H10

XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N

(f) Disturbanced7 = XN2

Figure 78 LNG production for different control structures as function of distur-bances for mode II Dots are re-optimized operation with constant composition


Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30

Figure 79∆Tsup as function of disturbance inWmaxs for the constantN control

strategy

optimized operation where bothN and∆msurge has been optimized The distur-bance is plotted on the X-axis with the nominal value at the center and the Y-axisshows the corresponding production rate in kgs-1 Note that the composition ofthe refrigerant is still the same as for the nominal operating point

N is actually the only feasible control structure for some disturbance directionsThis may be seen from Figure78(a) where all lines except the constantN lineends at the nominal point (from left to right) The reason for this is that theothercontrol structures would require a non-feasible rotational speed (higher than themaximum of 100) to maintain the controlled variable constant

From Figure78 we see that controllingN seems to be a good self-optimizingcontrol strategy that gives close to optimal operation over the entire disturbancerange This means that the implementation is quite simple sinceN = Nmax isoptimal in one region and close to optimal in the other regions

Note that we have notconsidered the constraint on∆Tsup in Figure 77 and inFigure78 The nominal value for∆Tsup is 129C (mode I) and 113C (mode II)which should be more than sufficient for normal operation For disturbances thatleads the process into operation in the region whereN = Nmax is optimal (eg tothe right in Figure78(a)) the super-heating is gradually reduced see Figure79but still remains positive At some point it will be necessary to discard one ofthe optimally active constraints (eg maximum shaft workWs = Wmax

s ) to satisfya minimum degree of super-heating One solution to this could be to select arefrigerant composition that would assure sufficient super-heating for all operatingpoints Another solution is to implement some logic that will switch controlledvariables depending on the operating point


N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85

Figure 710 LNG production as function ofN

In summaryFor both mode I and mode II fixing∆msurge= 0 andN = 100is optimal for the nominal operating point and in some of the disturbance regionsSince maintaining constantN and∆msurgeis also close to optimal for the remainingdisturbance regions we propose to use this control structure

78 Discussion

781 Moving temperature profile

An interesting result to note from Figure77and Figure78is the ldquokinkrdquo that seemsto be at the nominal point for all disturbances This was at first puzzling tous sincewe are not changing the set of constraints that are active

Let us consider the nominal point withN as degree of freedom (not worrying aboutthe constraintNmax which can be exceeded for a short time) Looking at the re-sulting graphs see Figure710 we note that ˙mLNG as a function ofN has a discon-tinuity at the nominal point (100) and one at approximately 105 We foundthat the reason behind this rather strange behaviour is the shape of the obtainedtemperature profile in the heat exchanger Figure76(b)shows that the tempera-ture difference profile in the heat exchanger has two clear peaks oneat the heatexchanger inlet and one about 70 through the heat exchanger Theldquokinkrdquo thatwe observe in the objective function ( ˙mLNG) occurs when the peak at the heat ex-changer inlet leaves the heat exchanger which causes an abrupt change in the heattransfer at the inlet to the heat exchanger Figure79 shows the effect in super-heating out of the heat exchanger

78 Discussion 121


The compressor characteristics used to link the flowrate pressure ratioefficiencyand the rotational speed of the compressor is not fitted to the data of a real com-pressor so it is uncertain if an installed compressor has the behaviour ourmodelpredicts An issue for further research could be to investigate if our conclusion onthe proposed control structure (∆msurge= 0kgs-1 andN = 100) is affected bythe compressor characteristic

783 Refrigerant composition

The refrigerant composition has been optimized for the nominal operating pointonly Since it is not adjusted in operation it would have been better to optimizeit with respect to the expected range of operating points (given by the distur-bances) This strategy ldquorobust optimumrdquo is discussed inGovatsmark and Sko-gestad(2005)

784 Additional considerations

The optimization problems for mode I and mode II presented above are simplifiedfor example

bull There might be a given schedule of ships arriving to transport the LNG tomarkets elsewhere Because of boil-off from the storage tanks it may bedesirable to minimize the storage but on the other hand there should besufficient storage to minimize the loading time of the ships This kind ofthinking is discussed byZaim(2002)

bull If there are more parallel production trains one needs to decide on howmany trains that should operate and how large production each train shouldhave This is also discussed byZaim(2002)

bull If the ambient temperature is changing (for example from night to day orseasonal changes) it may be optimal to produce more LNG during night(or in colder periods) when the gas turbine has higher efficiency and thecondenser temperature is lower Related topics are discussed for simplerrefrigeration systems inCai et al(2007) (also included as AppendixB)

122 Bibliography

Bibliography

Cai J Jensen J Skogestad S and Stoustrup J (2007) Balancing energy con-sumption and food quality loss in supermarket refrigeration systemin lsquoIEEErsquo


Govatsmark M and Skogestad S (2005) lsquoSelection of controlled variables androbust setpointsrsquoIndEngChemRes44(7) 2207ndash2217

Gravdahl J T and Egeland O (1999)Compressor Surge and Rotating StallModeling and Control London Springer Verlag



Moore F K and Greitzer E M (1986) lsquoA theory of post-stall transients in a axialcompressor systems Part I - Development of equationsrsquoJournal of Engineeringfor Gas Turbines and Power108 68ndash76


Saravanamuttoo H I H Rogers G F C and Cohen H (2001)Gas turbinetheory 5th edn Pearson Education Limited Harlow England

Singh A and Hovd M (2006) Dynamic modeling and control of the PRICOLNG processin lsquoAmerican Institute of Chemical Engineers (AIChE) annualmeetingrsquo


Skogestad S (2002) Plantwide control towards a systematic procedure in lsquoEu-ropean Symposium on Computer Aided Process Engineering (ESCAPE)- 12The Hague The Netherlandsrsquo

Stebbing R and OrsquoBrien J (1975) An updated report on the PRICO(TM) pro-

Bibliography 123

cess for LNG plantsin lsquoGASTECH LNG Natural Gas LPG international con-ference Parisrsquo

Zaim A (2002) Dynamic optimization of an LNG plant Case study GL2Z LNGplant in Arzew Algeria PhD thesis RWTH Aachen

Table 73 Structure of model equations

Unit Equations

Compressor Ws = mmiddot (houtminushin) Ws = m(hsminushin)ηPoutPin

= f (mr Nr) η = f (mr Nr)lowast

Turbines Ws = mmiddot (houtminushin) Ws = m(hsminushin)η

Valves hout = hin m= zmiddotCv middotradic

∆Pmiddotρ

SW cooler Q = mmiddot (houtminushin)

Heat exchanger Qi(z) = Ui middot∆Ti(z)δAi mmiddot parthi(z)partz = Qi(z)dagger

partPi(z)partz = constant

lowastConsult Section782for details regarding the compressor characteristicdaggeri is the stream either natural gas hot refrigerant or cold refrigerant

Table 74 Data for the PRICO processUA for natural gas in main heat exchanger 845MWC-1

UA for warm refrigerant in main heat exchanger 532MWC-1

UA for cold refrigerant in main heat exchanger 616MWC-1

Nominal compressor isentropic efficiency 822 lowast

Isentropic efficiency for the liquid turbines 800

lowastAt nominal conditions andN = 100 see Section782for further details

124 Bibliography

Table 75 Optimal operation with disturbances (nominal refrigerant composition)Mode I Mode II

Ws Ph Pl N ∆msurge mLNG Ph Pl N ∆msurge

[MW] [bar] [bar] [] [kgs-1] [kgs-1] [bar] [bar] [] [kgs-1]d1(minus1) 695 266 375 999 0489d1(minus05) 727 281 394 100 000d1(0) Not relevant for mode I 767 300 414 100 000d1(05) 785 316 434 100 000d1(1) 798 332 453 100 000d2(minus1) 115 297 402 100 0000 736 301 415 100 0000d2(minus05) 110 280 382 100 0000 752 301 415 999 0000d2(0) 106 268 367 100 0000 767 300 414 100 0000d2(05) 105 263 375 960 1125 778 299 416 994 0000d2(1) 104 258 359 100 0000 786 298 421 980 0000d3(minus1) 100 256 347 100 0000 798 309 418 100 0000d3(minus05) 103 261 357 100 0000 782 304 416 999 0000d3(0) 106 268 367 100 0000 767 300 414 100 0000d3(05) 111 279 388 100 0024 733 292 411 100 0000d3(1) 117 294 410 991 0000 708 287 409 100 0602d4(minus1) 110 293 391 100 0000 753 311 420 100 0000d4(minus05) 107 277 375 100 0000 762 306 417 100 0000d4(0) 106 268 367 100 0000 767 300 414 100 0000d4(05) 109 270 388 956 1832 749 290 409 100 1002d4(1) 111 270 380 100 0 738 285 407 100 0471d5(minus05) 106 270 369 100 0000 758 298 421 979 0000d5(0) 106 268 367 100 0000 767 300 414 100 0000d5(05) 108 271 372 100 0000 764 301 415 100 0000d5(1) Infeasible 757 303 415 100 0000d6(minus1) 112 267 379 100 0000 720 280 410 980 0000d6(minus05) 110 268 376 100 0000 743 288 416 976 0000d6(0) 106 268 367 100 0000 767 300 414 100 0000d6(05) 108 280 377 100 0000 757 308 426 979 0000d6(1) Infeasible 755 320 431 100 0000d7(minus1) Infeasible 757 302 416 100 0503d7(minus05) 108 273 374 100 0000 763 300 414 100 1971d7(0) 106 268 367 100 0000 767 300 414 100 0000d7(05) 106 267 372 986 0367 764 299 418 987 0000d7(1) 107 268 376 978 0657 756 297 425 963 0000d8(minus1) 103 254 353 100 0000d8(minus05) 104 260 370 962 2493d8(0) 106 268 367 100 0000 Not relevant for mode IId8(05) 111 282 385 100 0000d8(1) Infeasible

di( j) i is the disturbance number (see below) andj is the fraction of the full disturbance (egj = minus1 is the minimum value of the disturbance andj = 05 is in the middle of the nominal valueand the maximum value of the disturbance)

d1 = Wmaxs d5 = xC2H6

d2 = Pfeed d6 = xC4H10

d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8

Degrees of freedom forrefrigeration cycles

Submitted for publication in Industrial amp Engineering Chemistry Research

An important issue for optimal operation and plantwide control is tofind the degrees of freedom available for optimization A previouslypublished systematic approach to determine the steady state degrees offreedom is expanded to take into account the active charge asa possibledegree of freedom in cyclic processes Additional degrees of freedomrelated to composition of the circulating refrigerant are also discussedTwo LNG processes of current interest the C3MR LNG process fromAir Products and the MFC process developed by Statoil-LindeLNGTechnology Alliance are studied with respect to operational degrees offreedom

81 Introduction

This paper considers degrees of freedom for available for optimization of refriger-ation processesSkogestad(2000) points out that it is normally steady-state thateffects the plant economics so we will consider only steady-state operation Weare then interested in the steady-state degrees of freedom that affects the plant eco-nomics (objective functionJ) Nopt = NMV minusN0 whereNMV is the manipulatedvariables (valves etc) andN0 is the variables that does not affect the economics(eg liquid level in an open process)Skogestad(2002 2004) The number of ma-nipulated variables are usually quite easily obtain by counting the valves pumpsand other inputs to the process TheN0 variables that does not affect the plant

125

126 Degrees of freedom for refrigeration cycles

economics however requires detailed process overview and understanding A listof the potential degrees of freedom for some typical process units is given inSko-gestad(2002) with an updated version inAraujo et al(2007) We wish to extendthis list to also include degrees of freedom that are special for closed cycles

Glemmestad et al(1999) discuss degrees of freedom for heat exchanger networks(HEN) The review paper on plantwide control byLarsson and Skogestad(2000)discuss degrees of freedom as this is an important step in plantwide controlAmore recent study on degrees of freedom is that ofKonda et al(2006)

Processes for liquefaction of natural gas are very cost intensive and requires largeamounts of energy in operation It is therefore important that the plants arebothwell designed and later operated close to optimum also for changing conditionsThe optimal design of LNG processes has been studied extensively by severalcompanies such as Air Products Shell Phillips and Statoil-Linde LNG Technol-ogy Alliance It seems however that the subsequent operation of LNGplants hasbeen less studied at least in the open literature This is a bit surprising consid-ering the large throughputs which makes even small improvements economicallyattractive There are some publications regarding control of LNG plants (Mandler2000 Singh and Hovd 2006) but they consider the dynamic performance andcontrollability rather than the optimal steady-state operationZaim (2002) lookedinto dynamic optimization of a plant with several trains in parallel

Degrees of freedom for refrigeration processes are covered in thenext section andin Section83we apply the findings on some case studies including the two LNGprocesses

bull The propane pre-cooled mixed refrigerant (C3MR) process from AirProd-ucts

bull The mixed fluid cascade (MFC) process from Statoil-Linde LNG Technol-ogy Alliance

82 Degrees of freedom

An important issue in plantwide control is to find the degrees of freedom thatmay be used for optimization (Skogestad 2000) which in our case is equal to thenumber of steady-state degrees of freedomNss This is an important number forseveral reasons First it determines the degrees of freedom availablefor solvingthe optimization problem However more importantly in terms of operation itdetermines the number of steady-state controlled variables (Nss) that need to be

82 Degrees of freedom 127

Table 81 Potential operational degrees of freedom (Nmaxss ) for some typical pro-

cess unitsProcess unit Potential DOFFeed 1 (feedrate)Splitter number of exit streams - 1Mixer 0Compressor turbine pump 1 (work)Adiabatic flash tank 0lowast

Liquid phase reactor 1Gas phase reactor 0lowast

Heat exchanger 1 (bypass or flow)Column (excluding heat exchangers) 0lowast + number of side streams Valve 0lowast

Choke valve 1Each closed cycleActive charge (holdup fluid) 1dagger

Composition of fluid NCminus1Dagger

lowastPressure is normally assumed to be given by the surrounding processand is then not a degree offreedom However on must add one degree of freedom for each extra pressure that is independentlyset and which has a steady-effect (need a manipulated input not already counted eg a valve)

daggerThe active charge in the equipment is a potential degree of freedom but it may not be availablein some designs

DaggerNC is the number of components in the working fluid (refrigerant)

selected Optimal operation is normally implemented by keeping those variablesat constant setpoints

Rule (actual degrees of freedom)The numberNss of steady-state degrees of free-dom may be obtained by counting the number of manipulated variablesNMV andsubtracting the followingN0 variables (Skogestad 2002 2004)

bull manipulated variables with no effect on the costJ eg extra bypasses ofheat exchangers (only used to improve dynamic performance)

bull variables with no steady state effect that need to be controlled eg liquidholdups with no steady state effect

ThusNss= NMV minusN0


Potential (maximum) degrees of freedom

Based on this ruleSkogestad(2002) derived the potential number of degrees offreedomNmax

ss for some typical process units and an updated version was publishedby Araujo et al(2007) In Table81 the list is further updated to include also thepotential degrees of freedom for cyclic processes The additions areshown belowthe dotted line in the table First a valve should normally not be counted unlessitaffects a pressure that has a steady-state effect An example is a chokevalve whichis installed to lower the pressure and it has therefore been added explicitlyas adegree of freedom in the table In addition we potentially haveNCminus1 degrees offreedom related to the fluid composition in the cycle Finally the active chargeinthe cycle is a potential degree of freedom For example it may change the pressurelevel This is explained in more detail below

Many designs will have fewer actual degrees of freedom than given inTable81For example a heat exchanger will not have any degrees of freedomif all flowsare given and there is no bypass or other means of affecting the heat transfer egthe main exchanger in a LNG process Similarly one may not be able to adjust thetotal active charge or fluid composition in practice

Potential degrees of freedom1 Compressor2 Heat exchangers1 Choke valve

+ 1 ldquoActive chargerdquoNmax

ss = 5 degrees of freedomQC

QH

Wsz

Ph

Pl

Evaporator

Condenser

Figure 81 A simple refrigeration cycle withNmaxss = 5 potential degrees of free-

dom However if the drawing shows the actual process there will only beNss= 4degrees of freedom because there is no means of changing the active charge byfilling or removing refrigerant

ExampleFrom Table81 the simple cooling cycle shown in Figure81with purefluid (NC = 1) has five potential degrees of freedom (Nmax

ss ) related to one com-pressor two heat exchangers one choke valve and the active charge Exactly how


these degrees of freedom may be changed depends on the case For example fora compressor the actual manipulated variable (MV) may be the rotational speed orthe fraction of time the compressor is on (home or automotive installations) Theactive charge in Figure81 may be changed by filling or removing refrigerant tothe cycle but since this is not included the actual process does not have a degreeof freedom related to the active charge and thus has only 4 degrees of freedom(Nss)

Table 82 Actual degrees of freedom for refrigeration cyclesNss= NMV minusN0

Process unit Actual DOFEach MV (Valve heat exchanger compressor turbine etc) 1For each cycle subtract variables with no steady-state effect (N0)Pure fluidLiquid receivers exceeding the firstlowast -1Multi componentLiquid receiver exceeding theNC firstdagger -1

lowastThe first receiver is not subtracted in a closed cycle as this has a steadystate effectdaggerAssumes composition different in each of theNC first receivers otherwise the number of de-

grees of freedom is less

821 Remark on active charge or ldquofeedrdquo for closed cycles

The degree of freedom related to the total active charge is not obvious so we herediscuss this ldquoextrardquo degree of freedom that may occur in closed cyclesConsiderthe process shown in Figure82 where we have included an external tank with avalve for filling refrigerant into the closed cycle A temperature controller adjuststhe compressor speed to assure that the cooling load is constant The choke valveis a thermostatic expansion valve (TEV) that controls the degree of super-heatingat the evaporator outlet With the fan speeds for the two heat exchangersfixed (egat maximum) there are then one potential remaining degree of freedom related tothe active charge

First assume that the process is operated with just enough charge (refrigerant in theclosed cycle) to obtain saturation out of the condenser see Figure82(a) We thenopen the external valve for some time and then close it again to fill more refrigerantinto the cycle We then get to the operating point shown in Figure82(b)where theadded amount of refrigerant has accumulated in the condenser This follows sincethe evaporator will not change its holdup (significantly) due to the thermostaticexpansion valve that indirectly sets the area available for super-heating so the only


TC

Super-heatcontrol

(a) Saturation at condenser outlet

TC

Super-heatcontrol

(b) Sub-cooling at condenser outlet

Figure 82 A simple (not closed) refrigeration cycle The external fillingemptyingsystem illustrates the degree of freedom related to the active charge

place the extra refrigerant can go is to the condenser The increased charge inthe condenser will lead to a higher pressurePh which again leads to larger meantemperature difference and thus more heat transfer and the liquid at the outlet fromthe condenser will be sub-cooled as indicated in Figure82(b) It has been shownthat this may reduce the necessary compressor power in some cases (Jensen andSkogestad 2007b) but the main point here is to note that the charge in the systemhas an effect on the steady state operation

Remark 1 Note that Table82discussed below does not apply because the cycle in Figure82 is not closed

Remark 2 If we add a liquid receiver to the cycle in Figure82then we loose one degreeof freedom (as we have a level with no steady-state effect that needs to be controlled) Toregain this degree of freedom we (at least) would need to add another valve

822 Actual degrees of freedom for refrigerant cycles

For a closedcycle in order to adjust the active charge during operation we needa liquid receiver (variable holdup) in the cycle However for a pure refrigerantadding additional (two or more) receivers will not increase the number ofdegreesof freedom as the holdup has no steady-state effect and needs to be controlled(actually it may reduce the number unless we also add a valve for this purpose)In general for a multicomponent (mixed) refrigerant the holdup of theNC firsttanks dohave a steady-state effect provided the tanks have different compositions


z

QC

QH

Ws

Pl

Ph

(a) No extra choke valveNMV = 4 andNss= 4

z

QC

QH

Ws

Pl

Ph

Pm

(b) An extra choke valveNMV = 5 andNss= 5

z

QC

QH

Ws

Pl

Ph

Pm

(c) Extra choke valve and liquid receiver on lowpressure sideNMV = 5 butNss= 4

Figure 83 Simple cycle with liquid receiver on the high pressure side


as they provide an indirect means of adjusting the fluid composition Only liquidreceivers exceeding theNC first will have no steady-state effect These argumentsare the basis for Table82which gives the actual(rather than the potentialin Table81) degrees of freedom for vapour compression cycles

Let us apply Table82 to the process in Figure81 which hasNMV = 4 (two heatexchangers one valve and one compressor) Since there are no liquidreceiversthere is no variables that need to be subtracted(N0 = 0) ThusNss= 4 degrees offreedom which confirms our earlier findings

Note that adding a liquid receiver somewhere in this cycle will not change thenumber of steady-state degrees of freedom In order to do this we also need to adda valve for example upstream of the receiver The addition of a liquid receiver tothe cycle is shown in Figure83 We have from Table82the following degrees offreedom for the three cases in Figure83

Figure 83(a)NMV = 4-N0 = 0Nss = 4

Figure 83(b)NMV = 5-N0 = 0Nss = 5

Figure 83(c)NMV = 5-N0 = 1Nss = 4

For the two first cases the liquid level does not need to be subtracted as ithas asteady-state effect (see Figure82) and also does not need to be controlled in aclosed cycle

The design in Figure83(a)with no extra valve does not allow for adjusting theactive charge The design in Figure83(b)with an extra choke valve has an ad-ditional degree of freedom This may be more optimal (Jensen and Skogestad2007b) since it allows for the condenser outlet to be sub-cooled This is becausethe pressure in the receiver is equal to the saturation pressure and the extra valvegivesP gt Psat (sub-cooling) at the condenser outlet

In Figure83(c) we have added also a liquid receiver on the low pressure sideThus we have lost one degree of freedom compared to Figure83(b) because oneof the liquid levels need to be controlled Another way of understanding whythereis one less degree of freedom is that we now always have saturated vapour at theinlet to the compressor whereas it before could be super-heated

Remark Note that a loss of a degree of freedom does not mean that the process is lessoptimal In fact in this case it is opposite because for a simple cycle (without internalheat exchange) it is optimal to have saturation (no super-heating) before the compressorThus Figure83(c)is optimal by design whereas in Figure83(b)one needs to adjust oneof the degrees of freedom to get optimality and this may be difficult to achieve in practice

83 Case studies 133

83 Case studies

We will here present some more complex case studies First we will look at twoprocesses not directly related to LNG plants a two-pressure level refrigeration cy-cle cooling a process stream and a heat integrated distillation column (two differentconfigurations) Then we will look at three LNG case studies a small scaleLNGprocess with a single mixed refrigerant the propane pre-cooled mixed refrigerant(C3MR) process from Air Products and the mixed fluid cascade (MFC) processfrom Statoil-Linde LNG Technology Alliance

831 Two-pressure level refrigeration

Figure84(a)shows a refrigeration system with two-stage expansion using a purerefrigerant (NC = 1) The process stream is first cooled by the intermediate-pressure refrigerant (Evaporator 1) and then by the low-pressurerefrigerant (Evap-orator 2) The evaporators are kettle type boilers so there is no super-heating of thevapour The temperature profile in the evaporators is illustrated in Figure84(b)Control of such cycles is discussed byWilson and Jones(1994) The two com-pressors are usually driven with a common driver so there is only one manipulatedvariable for the two compressors1copy There are three valves2copy 3copy and 4copy shownin Figure84(a) The valve between Evaporator 1 and the second compressor4copyis present to limit the amount of cooling in Evaporator 1 if necessary In additionwe may manipulate the flow of coolant in the condenser5copy and the process stream6copy Thus this two-pressure level cycle hasNMV = 6 manipulated variables

The two evaporators in Figure84(a)will function as liquid receivers so there are intotal three variable liquid levels Two of the liquid levels typically the evaporatorsneed to be controlled for stabilization and since the refrigerant is pure thesetpointsfor these (levels) has no steady-state effect (see also Table82) Thus we end upwith Nss= NMV minusN0 = 6minus2 = 4 degrees of freedom To operate the system weneed to decide on four controlled variables In general these should beselectedas the possible active constraints (eg max cooling5copy fully open) plus ldquoself-optimizingrdquo variables

It is interesting to compare the actualdegrees of freedom with the potentialdegreesof freedom according to Table81 We have for Figure84(a)


Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2


Figure 84 Cooling at two pressure levelsNss= 4

Actual DOFNMV = 6minusN0 = 2Nss = 4

Potential DOF1 Feed1 Compressors3 Heat exchangers3 Valve

+ 1 Active chargeNmax

ss = 9 degrees of freedom

The 5 ldquolostrdquo degrees of freedom are related to

1 No sub-cooling in the condenser (not optimal)

23 No super-heating in the two evaporators (optimal)

45 No bypass for the two evaporators (optimal)

Thus four of the five lost degrees of freedom are ldquooptimal by designrdquoin this caseThe only possible loss is related to not allowing for sub-cooling of the streamleaving the condenser To fix this would require an additional valve between thecondenser and the liquid receiver

832 Heat integrated distillation

To reduce the energy consumption in distillation one may use a heat pump be-tween the condenser and reboiler (Salim et al 1991) Two possible designs are

83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy

(a) External working fluid

1copy

2copy

3copy

4copy

5copy6copy

7copy

(b) Vapour overhead as working fluid

Figure 85 Two ways of using a heat pump to integrate the reboiler with the con-denser in a distillation column

shown in Figure85

a) With external working fluid Control of such columns have been studiedbyJoslashrgenses and coworkers (Hallager et al 1990 Nielsen et al 1987 1988Li et al 2003) who also have an experimental setup

b) With the vapour overhead as working fluid (not a closed cycle)

There are in total 7 manipulated variables for both systems (see Figure85) Feedflowrate 1copy Bottom product flowrate2copy Top product flowrate3copy Reflux flowrate4copy Compressor5copy Choke valve6copy Cooling water flowrate7copy

On the column side there are two liquid levels with no steady-state effect that mustbe controlled

bull Condenser liquid level (eg may be controlled by the reflux flowrate4copy)

bull Reboiler liquid level (eg may be controlled by the bottom product flowrate2copy)

Using the method in Table82 combined with the general rule we get the actualdegrees of freedom


Figure 85(a)NMV = 7minusN0 = 3lowast

Nss = 4

lowastOne of the two levels in the heat pumpcycle must be controlled and has no steady-state effect

Figure 85(b)NMV = 7minusN0 = 3lowast

Nss = 4

lowastThe heat pump cycle is no longer closedso the single liquid level (receiver) must becontrolled

Thus in both cases there are 4 steady-state degrees of freedom whenthe feedrateand column pressure are included This is the same as for an ldquoordinaryrdquo distillationcolumn

Govatsmark(2003) studied the process with external working fluid (Figure85(a))and found that the top composition and the column pressure are at their constraintsfor the case where the top product is the valuable product Assuming a given feedflowrate there is then one unconstrained degree of freedom left This unconstraineddegree of freedom could then be used to control a temperature in the bottomsectionof the column which is found to be a good self-optimizing controlled variable(Govatsmark 2003)

It is interesting to compare the actualdegrees of freedom with the potential(max-imum) degrees of freedom according to Table81

Figure 85(a)1 Feed0 Column1 Pressure (in column)1 Compressor3 Heat exchanger1 Choke valve



Figure 85(b)1 Feed0 Column1 Pressure (in column)1 Compressor2 Heat exchanger1 Choke valve

+ 1 Pressure (in condenser)Nmax


In Figure85(a)the four ldquolostrdquo degrees of freedom are related to

12 No bypass of two heat exchangerslowast (optimal)

3 Saturation before compressor (optimal)

4 Saturation at condenser outlet (not optimal)

For Figure85(b)there are three ldquolostrdquo degree of freedom related to

1 No bypass of the column reboiler (optimal)

lowastNot the cooler because the flow of coolant is an actual degree of freedom

83 Case studies 137



The only non-optimal ldquolostrdquo degree of freedom is for both cases related tosatu-ration out of the condenser in the heat pump cycle This may be close to optimal(Jensen and Skogestad 2007a) However to gain this degree of freedom it is nec-essary to have a valve between the cooler and the liquid receiver This valve willthen give sub-cooling out of the cooler

833 Small scale LNG process

From Table 82NMV = 6minusN0 = 0lowast

Nss = 6

lowastNo subtraction of liquid receivers be-cause of multicomponent working fluid withNCge 2

1copy

2copy 3copy

4copy

5copy

6copy NG

LNG

Figure 86 A small scale LNG concept to illustrate the degrees of freedom relatedto changing the composition via liquid levels

Consider the small-scale LNG process in Figure86 with a mixed refrigerantNCge 2 (Neeraas and Brendeng 2001) Note that the original process design hasadditional degrees of freedom related to individual heat exchangersfor refriger-ant cooling and natural gas cooling Our simplified flowsheet assumes thatit is


optimal to cool the refrigerant and the natural gas to the same temperature Wefind (using Table82) that the process has six manipulated variablesNMV = 6 seeFigure86 Since there are two liquid levels we need to control at least one forstabilization Thus it is tempting to remove one degree of freedom Howeverthelevel setpoint has a steady-state effect because the composition in the two tanksare different and we may change the composition of the circulating refrigerant byshifting mass from one tank to the other ThusNss= NMV = 6

Using Table81we get the potential degrees of freedom if we assumeNC = 3

Figure 861 Feed1 Compressor3 Choke valve4 Heat exchanger1 Active charge2 Compositions

Nmaxss = 12 degrees of freedom

We have the following 6 lost degrees of freedom

1-3 No bypass of process heat exchangers (optimal)

45 Pressure in the two flash drums (optimal discussed below)

6 Composition of the refrigerant (not optimal)

The ldquolostrdquo degree of freedom related to the pressure in the two flash drumsare notobvious Adding a valve before the second flash drum will give a lower tempera-ture in the flash drum and thus also of the vapour that is sent through the secondheat exchanger This is not optimal since the vapour will then be colder than thenatural gas which it is cooled together with The same is true for the first flashdrum if the natural gas feed is cooled with the same coolant as the refrigerant aftercompression

834 Propane pre-cooled mixed refrigerant (C3MR)

The C3MR process developed by the Air Products company has a large marketshare of the existing liquefaction plants worldwide A flowsheet of the C3MRprocess is given in Figure87 The first cycle is with a pure refrigerant usuallypropane The second cycle is with a mixed refrigerant We identify the followingmanipulated variables

bull Natural gas feed1copy

83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy

Figure 87 Flowsheet of the C3MR processPropane (C3) Mixed refrigerant(MR) Natural gas (NG)

bull 6 choke valves for propane pre-cooling (one for each pressure level for natu-ral gas cooling and one for each pressure level for mixed refrigerant cooling)2copy 3copy 4copy 5copy 6copy and 7copy

bull Propane compressor one speed8copy

bull Flow of cooling water or air in propane condenser9copy

bull Two choke valves for mixed refrigerant cycle10copy and11copy

bull Mixed refrigerant compressor12copy

bull Flow of cooling water or air in mixed refrigerant cooler13copy

For the propane cycle we need to control 6 of the 7 liquid levels (eg the heatexchanger levels) and sinceNC = 1 none of these level setpoints will have a steady-state effect AssumingNC = 3 for the mixed refrigerant cycle we get the followingactual and potential degrees of freedom


Actual DOFNMV = 13minusN0 = minus6Nss = 7

Potential DOF1 Feed2 Compressor10 Heat exchanger8 Choke valve2 Active charge

+ 2 Compositions (MR)Nmax




9-14 Saturation at propane compressor inlet (no super-heating optimal)

15 Saturation out of propane condenser (no sub-cooling not optimal)

16 Pressure in the flash drum in the MR cycle (optimal)

1718 Composition of the mixed refrigerant (not optimal)

The only loss in efficiency due to the ldquolostrdquo degrees of freedom are caused by hav-ing no sub-cooling in the propane condenser and by not being able to change thecomposition in the mixed refrigerant cycle To get sub-cooling in the condenser itis necessary to have a valve between the condenser and the liquid receiver Adjust-ing the composition is discussed further in Section84

It is optimal to have the flash drum in the mixed refrigerant cycle at the samepressure as the outlet from the last propane cooler Otherwise the warmrefrigerantwould be colder than the natural gas in the first mixed refrigerant heat exchangercausing a non-optimal temperature profile

Optimal operationLet us consider two different operating strategies

Case 81Maximum production given available shaft work

Case 82Minimum shaft work given feed flowrate

For both cases the following is true

It is not economical to cool more than necessary so the natural gas outlettemper-ature is at its maximum constraintlowast this may be controlled by the last choke valve

lowastThis temperature will implicitly set the amount of flash gas see Chapter6 and the compositionof both the flash gas and the LNG

83 Case studies 141

in the mixed refrigerant cyclelowast removing one degree of freedom Cooling wateris usually cheap so it is usually wise to maximize the flow of cooling water whichremoves two additional degrees of freedom We are then left with 4 degrees offreedom to optimize the operation

For Case81 Two degrees of freedom are used to maximize compressor shaftwork one for each cycle (active constraints) This leaves us with two uncon-strained degrees of freedom so two setpoints must be specified for this case (egPh in the mixed refrigerant cycle andPl in the propane cycle)

For Case82 The feed flowrate is given so we loose one degree of freedom Thisleaves us with three unconstrained degrees of freedom so three setpoints must bespecified for this case (egPl andPh in the mixed refrigerant cycle andPl in thepropane cycle)

Remark If the cooling water is too cold it may be necessary to limit the flow of coolingwater to avoid violating constraints on the compressor suction pressure (minimum con-straint) This however does not change the analysis sincethe number of active constraintsare the same since we exchange the maximum cooling constraint with the minimum pres-sure constraint

835 Mixed fluid cascade (MFC)

The Statoil-Linde LNG Technology Alliance has developed a mixed fluid cascade(MFC) process (Bach 2002 Forg et al 1999) A flowsheet of the process is givenin Figure88 It consists of three refrigeration cycles i) Pre-cooling cycle (PR) ii)liquefaction cycle (LC) and iii) sub-cooling cycle (SC) All three refrigerant cyclesuse mixed refrigerants

There are in total 13 manipulated variables (NMV = 13)


bull Flow of cooling waterair in PC cycle2copy

bull Choke valve intermediate pressure level PC cycle3copy

bull Choke valve low pressure level PC cycle4copy

bull Compressor PC cycle5copy

bull Flow of cooling waterair in LC cycle6copy

lowastAnother solution is to use the natural gas feed flowrate by adjusting the LNG expansion Thisexpansion device (valve or turbine) is not shown here as we have indicated the feed manipulator atthe inlet of the stream instead


1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG

Figure 88 Flowsheet of the MFC processPre-cooling cycle (PC) Liquefactioncycle (LC) Sub-cooling cycle (SC)andNatural gas (NG)

bull Extra valve LC cycle7copy

bull Choke valve LC cycle8copy

bull Compressor LC cycle9copy

bull Flow of cooling waterair in SC cycle10copy

bull Extra valve SC cycle11copy

bull Choke valve SC cycle12copy

bull Compressor SC cycle13copy

There are no liquid levels that must be controlled (N0 = 0) AssumingNC = 3 foreach cycle we get the following actual and potential degrees of freedom

84 Discussion 143


Potential DOF1 Feed3 Compressor7 Heat exchangers4 Choke valve3 Active charge

+ 6 CompositionNmax




5 Saturation at PC condenser outlet (not optimal)

6-11 Two compositions for each of the three cycles (not optimal)

Also for this process it is the saturation specification out of the condenserfor thefirst cycle and the fixed compositions that will give the losses

Optimal operation Let us again consider optimal operation for two cases TheLNG outlet temperature must be controlled and the amount of cooling water inthree sea water coolers are maximized giving 9 unconstrained degrees of freedom

For Case81(maximum feed)We use three degrees of freedom to maximize thecompressor shaft work in each cycle We are then left with 6 unconstrained degreesof freedom so 6 setpoints must be specified

For Case82(given feed)The feed is given so there are then 8 unconstrained de-grees of freedom For this case we need 8 setpoints

Note that there may be active constraints on the temperature after PCHX1 andorthe temperature after PCHX2 if the NGL extraction is integrated For each activeconstraint we will have one less unconstrained degree of freedom thatwe have tofind a controlled variable for

84 Discussion


Table 81 is not straightforward to use in practice This is mainly because anintermediate pressure may or may not have a steady-state effect This is noteasilycaptured by simply counting the degrees of freedom for each unit operation but


requires a process understanding The method shown in Table81 is only one ofseveral alternatives that will lead to the same result


For mixed refrigerants there areNC minus 1 potential degrees of freedom related tothe composition of the refrigerant However all of these are usually not realizedas actual degrees of freedom We have claimed above that liquid level setpoints(provided different composition) may be used to effectively utilize the degreesof freedom related to refrigerant composition see the small scale LNG processpresented above This is not a practical solution with several compositionsas it isnecessary with equally many liquid tanks (with sufficiently different compositions)and control elements (valves) So in practice one will instead rely on a constantcomposition that may be changed on a larger timescale by utilizing the make-upsystem

843 Saturation in condenser

Another potential degree of freedom that is sometimes ldquolostrdquo is related to the con-denser pressure By having a liquid receiver after the condenser it will not be possi-ble to have a condenser pressure different from the saturation pressure (Pcon= Psat)By having a valve in between the condenser and the liquid receiver it is possibleto have sub-cooling in the condenser (Pconge Psat) This is discussed in more detailin Jensen and Skogestad(2007b) However it may also be necessary with a dif-ferent condenser design if the design does not allow for sub-coolingThis is thecase if the liquid formed in the condenser leaves the heat transfer zone (eg due togravity)

85 Conclusion

The degrees of freedom available for optimization (Nss) is an important numberfor several reasons It determines the number of free variables available to solvethe optimization problem However more importantly it determines how manysteady-state controlled variables that must be selected to operate the process

This paper extends an earlier published simple approach to determine the potentialdegrees of freedom (Nmax

ss ) based on unit operations to also cover vapour compres-sion cycles A simple method to determine the actualdegrees of freedom (Nss)

Bibliography 145

for vapour compression cycles is also presented Both methods are illustrated onfour case studies where the difference between the potentialand actualdegrees offreedom are explained and related to the process layout

For the two LNG case studies (C3MR and MFC) we also illustrate the effect ofoperating strategy (maximum production and given production) on the number ofunconstrained degrees of freedom

Bibliography

Araujo A Govatsmark M and Skogestad S (2007) lsquoApplication of plantwidecontrol to the HDA process I - steady-state optimization and self-optimizingcontrolrsquo Control engineering practice15 1222ndash1237

Bach W A (2002) lsquoDevelopments in the mixed fluid cascade process (MFCP)for LNG baseload plantsrsquoReports on science and technology Linde63

Forg W Bach W Stockmann R Heiersted R S Paurola P and FredheimA O (1999) lsquoA new LNG baseload process and manufacturing of the mainheat exchangerrsquoReports on science and technology Linde61

Glemmestad B Skogestad S and Gundersen T (1999) lsquoOptimal operation ofheat exchanger networksrsquoComput Chem Eng23 509ndash522

Govatsmark M S (2003) Integrated Optimization and Control PhD thesisNor-wegian University of Science and Technology (NTNU) Trondheim Norway

Hallager L Jensen N and Joslash rgensen S B (1990) Control system configura-tion for a heat pump operating as energy source for a distillation columninlsquoPresented at Nordic CACE symposium Lynby Denmarkrsquo



Konda N Rangaiah G and Krishnaswamy P (2006) lsquoA simple and effectiveprocedure for control degrees of freedomrsquoChem Engng Sci61 1184ndash1194

Larsson T and Skogestad S (2000) lsquoPlantwide control - a reviewand a newdesign procedurersquoModeling Identification and Control21 209ndash240

146 Bibliography

Li H Gani R and Joslash rgensen S B (2003) Integration of design and controlfor energy integrated distillationin lsquoEuropean Symposium on Computer AidedProcess Engineering (ESCAPE) 13 Lappeenranta Finlandrsquo

Mandler J A (2000) lsquoModelling for control analysis and design in complexindustrial separation and liquefaction processesrsquoJournal of Process Control10 167ndash175

Neeraas B O and Brendeng E (2001) lsquoMethod and device for small scale liq-uefaction of a product gasrsquo US Patent number 6751984

Nielsen C S Andersen H W Brabrand H and Joslashrgensen S B(1988) Adap-tive dual composition control of a binary distillation column with a heat pumpin lsquoPreprints IFAC Symp on Adaptive Control of Chem Processesrsquo

Nielsen C S Andersen H W Brabrand H Toftegard B and Joslashrgensen S B(1987) Dynamics and identification of distillation pilot plant with heat pumpin lsquoPresented at CHISArsquo87 Praguersquo pp 1ndash10

Salim M A Sadasivam M and Balakrishnan A R (1991) lsquoTransient analysisof heat pump assisted distillation system 1 the heat pumprsquoInt J of EnergyResearch15 123ndash135




Skogestad S (2004) lsquoNear-optimal operation by self-optimizing control Fromprocess control to marathon running and business systemsrsquoComput Chem Eng29 127ndash137



Chapter 9

Conclusion

The charge (holdup) does not affect the steady-state for anopenprocess (eg liquidlevel in a buffer tank) because of the boundary conditions on pressure It has beenshown that ldquoactive chargerdquo inclosedcycles has a steady-state effect One wayof affecting the ldquoactive chargerdquo is by having a liquid receiver in the cycle Thisdegree of freedom is often lost by designing the cycle without sub-cooling in thecondenser We find that some sub-cooling is desirable and for the ammonia casestudy the compressor shaft work is reduced by about 2 by allowing forsub-cooling The savings are not very large but more importantly the results showthat the active charge is a degree of freedom and that the sub-cooling gives somedecoupling between the high pressurePh and the hot source temperatureTH Thisshows that there are no fundamental differences between the typical sub-criticalcycles and the trans-criticalCO2 cycles

In terms of practical operation there are differences between the sub-critical am-monia cycle and the trans-criticalCO2 cycle For the ammonia cycle several simplecontrol structures gives acceptable performance The best controlstructure foundis to control the temperature approach at the condenser exit For theCO2 cycle wehad to use a linear combination of measurements to get acceptable performance

It is common to do the early design of refrigeration cycles by specifying a mini-mum approach temperature in the heat exchangers (minJ subject to∆T ge ∆Tmin)This method fails to give the optimal operating point and also fails to find that sub-cooling is optimal As a simple alternative we propose the simplified TAC method(min(J+C0 sumAn

i )) whereC0 replaces∆Tmin as the adjustable parameter

A PRICO LNG process has been designed by using realistic compressor specifi-cations found online We are able to increase the LNG production comparedwith

147

148 Conclusion

the commercial PRICO process Compared with the AP-X LNG process we findthat the PRICO process has about 7 less production for the same available shaftpower

Operation of the PRICO process is studied for two modes of operation i) minimumshaft work (for given production) and ii) maximum production Both modeshas 2controlled variables that must be selected (after controlling constraints thatare al-ways optimally) We find that maintaining a minimum distance to surge (∆msurge=00kgs-1) and maximum rotational speed of the compressor (N = 100) gives op-timal operation for the nominal operating point and for some of the disturbances(for both modes) Since this control structure also gives close to optimal opera-tion for the remaining disturbances we propose to control∆msurge= 00kgs-1 andN = 100

The degrees of freedom available for optimization is an important number forsev-eral reasons First it determines the number of free variables to solve theoptimiza-tion problem Second it determines how many steady-state controlled variablesthat must be selected to operate the process Finding the degrees of freedom isnot straightforward and requires a detailed process understanding An earlier pub-lished simple approach to determine the potential degrees of freedom basedon unitoperations is extended to also cover vapour compression cycles A simple methodto determine the actual degrees of freedom for vapour compression cycles is alsopresented Both the methods are illustrated on four case studies where the differ-ence between potential and actual degrees of freedom are explained and related tothe process layout

Appendix A

Optimal operation of a mixedfluid cascade LNG plant

Studies on the operation of complex vapour compression cycles likethe one used for the production of liquefied natural gas (LNG) are notwidely reported in the open literature This is a bit surprising consider-ing the large amount of work that has been put into optimizingthe designof such processes It is important that the process is operated close tooptimum to fully achieve the maximum performance in practice Thereare possibilities for savings both due to (a) identifying the optimal pointof operation and (b) selecting the controlled variables such that the op-timal operation depends weakly on disturbancesIn this paper we study the mixed fluid cascade (MFC) LNG process de-veloped byThe Statoil Linde Technology Alliance We study the degreesof freedom and how to adjust these to achieve optimal steady-state op-eration

A1 Introduction

Large amounts of natural gas (NG) are found at locations that makes it infeasible ornot economical to transport it in gaseous state (in pipelines or as compressed NG)to the customers The most economic way of transporting NG over long distancesis to first produce liquefied natural gas (LNG) and then transport the LNG by shipsAt atmospheric pressure LNG has approximately 600 times the density of gaseousNG

At atmospheric pressure LNG has a temperature of approximatelyminus162C so

149

150 Optimal operation of a mixed fluid cascade LNG plant

the process of cooling and condensing the NG requires large amounts of energySeveral different process designs are used and they can be grouped roughly asfollows

bull Mixed refrigerant The refrigerant composition is adjusted to match thecooling curve of NG Some are designed with a separate pre-cooling cycle

bull Cascade process (pure fluid) Several refrigerant cycles are used to limit themean temperature difference in the heat exchange

bull Mixed fluid cascade process Energy efficiency is further improved byusingseveral mixed refrigerant cycles

The process considered in this paper is the Mixed Fluid Cascade (MFC) processdeveloped byThe Statoil Linde Technology Alliance(Bach 2002) The MFC pro-cess has three different cycles all with mixed refrigerant and the firstcycle withtwo pressure levels

The steady-state model for this plant is implemented in gPROMS (nd) result-ing in approximately 14000 equations Optimizing the plant takes in the orderof 2 hours on a Pentium 4 computer with 28 GHz and 512 MB RAM runningGNULinux

A2 Process description

A simplified flowsheet is given in FigureA1 For more details about the processconsultBach(2002) andForg et al(1999)

Nominal conditions

bull Feed NG enters withP = 615bar andT = 11C after pretreatment Thecomposition is 888 methane 57 ethane 275 propane and 275nitrogen Nominal flow rate is 1kmols-1

bull Product LNG is atP = 551bar andT = minus155C

bull The refrigerants are a mix of nitrogen (N2) methane (C1) ethane (C2) andpropane (C3) and the compositions are used in optimization

bull The refrigerant vapour to the compressors are super-heated 10C

bull The refrigerants are cooled to 11C in all sea water (SW) coolers (assumedmaximum cooling)

A3 Degree of freedom analysis 151

NG

LC

SC

PC1 PC2

LNG

PCHX1 PCHX2 LCHX SCHX

Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW

Figure A1 Simplified flowsheet of the MFC process SC - sub-cooling cycleLC - liquefaction cycle PC - pre-cooling cycle (two stages 1 and 2) Degrees offreedom associated with variable active charge in each cycle are not shown

bull Pressure drops are 05bar in SW coolers 05bar for hot flows in main heatexchangers and 02bar for cold refrigerant in main heat exchangers

The SRK equation of state is used both for NG and the refrigerants The heatexchangers are distributed models with constant heat transfer coefficients Thecompressors are isentropic with 90 constant efficiencies

A3 Degree of freedom analysis

In this section we present a detailed degree of freedom analysis which is an impor-tant result of this work

In a single simple vapour compression cycle (eg a home refrigerator) there aretwo obvious manipulated inputs namely the compressor and the valvelowast In addi-tion there is a less obvious manipulated variable This is the ldquoactive chargerdquoin thecycle which may be modified by introducing a unit (tank) with variable holdup(Jensen and Skogestad 2007) The active charge may be changed by placing tanks

lowastIn addition one might control flow of hot and cold fluid but this is outside thecycle so let usoverlook that for now


at many different locations but from a simple mass balance it may be verifiedthatfor each cycle one may have only one independent variable (tank) associated withthe active charge Thus for the cycles the number of manipulated variablesare thenumber of compressors and valves plus one active charge for each cycle

Let us now look at the MFC process

A31 Manipulated variables (MVrsquos)

From the discussion above we find that there are in total 26 manipulated variables(degrees of freedom)

bull 5 Compressor powersWsi

bull 4 Choke valve openingszi

bull 4 SW flows in coolers

bull 1 NG flow (can also be considered a disturbance)

bull 9 Composition of three refrigerants

bull 3 active charges (one for each cycle)

A32 Constraints during operation

There are some constraints that must be satisfied during operation

bull Super-heating The vapour entering the compressors must bege 10C super-heated

bull ToutLNG NG Temperature out of SCHX must beleminus155C or colder

bull Pressure 2barge Ple 60bar

bull NG temperature after PCHX1 and PCHX2 (not considered in this paper)

bull Compressor outlet temperature (not considered in this paper)

A33 Active constraints

We are able to identify some constraints that will be active at optimum In totalthere are 11 active constraints

A4 Optimization results 153

bull Excess cooling is costly soToutLNG = minus155C

bull Optimal with low pressure in cycles soPl = 2bar (for all 3 cycles)

bull Maximum cooling AssumeT = 11C at 4 locations

A34 Unconstrained degrees of freedom

After using 11 of the 26 manipulated inputs to satisfy active constraints we areleft with 15 unconstrained degrees of freedom In this work we considerthe NGflow given from elsewhere (disturbance to the process) In addition weassumethat the degree of super-heating is controlled at∆Tsup= 10C so we are left with13 degrees of freedom in optimization For a steady state analysis the pairingofinputs and outputs is insignificant so say we are left with the following subsetofthe MVrsquos

bull 3 NG temperatures (after PCHX1 PCHX2 and LCHX)

bull Pm in SC

bull 9 Refrigerant compositions

In this paper we will not consider manipulating refrigerant composition in opera-tion (only in the optimization) so of the 13 unconstrained degrees of freedom weare left with 4 during operation

A4 Optimization results

In this section we are optimizing on the 13 degrees of freedom given abovetolocate the optimal operation of a given MFC LNG plant The resulting temperatureprofiles for the four main heat exchangers are given in FigureA2 Some key valuesof the refrigerant cycles are given in TableA1 where the nomenclature is given inFigureA1

Some remarks

bull The total shaft work is 10896MW

bull The optimal NG temperature out of PCHX1 PCHX2 and LCHX isminus173Cminus515C andminus771C respectively


middot - middot LIQ- - - PREmdash- NG SUBmdash- PREcold

Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A

middot - middot LIQ- - - PREmdash NG SUBmdash PREcold

0 50 100 150215220225230235240245250255260

(b) NG1B

mdash NGmiddot - middot LIQ SUBmdash LIQcold

Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2

mdash NG SUBmdash SUBcold

Position0 50 100 150

120130140150160170180190200

(d) NG3

Figure A2 Temperature profiles

bull In the true design there will separators at the high pressure side of the cycleswhich has not been considered here Further work will include an analysisof the effect of this sub-optimal design

bull In the SC cycle the pressure ratios over the two compressor stages are farfrom equal (which is a rule of thumb for compression ratios) This is becausethe inlet temperature to the first stage (approximatelyminus80C) is much lowerthan inlet temperature to the second stage (11C)

bull Nitrogen is present in SC only to satisfy the minimum pressure of 2bar

A5 Control structure design 155

Table A1 Optimal operation of a MFC processPC1 PC2 LC SC

Pl [bar] 645 200 200 200Pm[bar] 645 - 2838Ph [bar] 1503 1503 2058 5699C1 [] 000 000 402 5299C2 [] 3770 3770 8296 4245C3 [] 6230 6230 1302 000N2 [] 000 000 000 455n[mols-1] 464 685 390 627Ws[MW] 12565 + 2644 2128 3780+1086

A5 Control structure design

NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH

Figure A3 Suggested control structure for the MFC process SH is degree ofsuper-heating controllers PC and TC are pressure and temperature controllers re-spectively Not shown Three pressure controllers on the low pressure side usingthe active charge in each cycle

In the section above we where able to identify the optimum for the process buthow should this optimum be implemented in practice First we need to control theactive constraints


bull Pl is for each of the 3 cycles For this we may use ldquoactive chargerdquo (seediscussion above)

bull Maximum cooling in 4 SW coolers SW flow at maximum

bull LNG outlet temperature atminus155C May use first compressor stage in SUB

In addition we choose to control

bull Degree of super-heating (4 locations) For this we may use the correspond-ing choke valve opening

The four remaining degrees of freedom should be used to control variables whichhave good self optimizing properties

ldquoSelf optimizing control is when we can achieve acceptable loss with constantsetpoint values for the controlled variables (without the need to re-optimize whendisturbances occur)rdquo (Skogestad 2000)

To evaluate the loss one needs to consider the effect of disturbances and implemen-tation errors A steady-state analysis is usually sufficient because the economicsare primarily determined by the steady-state

Based on physical insight the following four variables may been suggested

bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm

A possible control structure with these four variables and the active constraintscontrolled is shown in FigureA3 However note that the ldquopairingsrdquo of controlledand manipulated inputs are included primarily to illustrate that we have availabledegrees of freedom as this does not matter for evaluating self-optimizing controlat steady-state It will be the subject of future work to compare this choiceofcontrolled variables with one that follows from a systematic procedure

A6 Conclusion

We have shown that the degrees of freedom in vapour compression cycles are equalto the number of compressors and valves plus one The extra degree of freedom is

Bibliography 157

related to the ldquoactive chargerdquo in the system and a tank with variable holdup shouldbe included to gain this degree of freedom

A detailed degree of freedom analysis for the MFC process reveals thatthere arefour unconstrained degrees of freedom in operation (not considering manipulatingrefrigerant compositions) To fully achieve the potentially high thermodynamicefficiency of the MFC process it is important that these four unconstrained degreesof freedom are utilized optimally

Bibliography

(nd)httpwwwpsenterprisecomproducts_gpromshtml





158 Bibliography

Appendix B

On the Trade-off between EnergyConsumption and Food QualityLoss in SupermarketRefrigeration Systems

J Cailowast J B Jensendagger S Skogestaddagger J Stoustruplowast

Submitted for publication in the proceedings of the 2008 American Control Con-ference Seattle USA

This paper studies the trade-off between energy consumption and foodquality loss at varying ambient conditions in supermarket refrigerationsystems Compared with the traditional operation with pressure controla large potential for energy savings without extra loss of food quality isdemonstrated We also show that by utilizing the relativelyslow dynam-ics of the food temperature compared with the air temperature we areable to further lower both the energy consumption and the peak valueof power requirement The Pareto optimal curve is found by off-lineoptimization

lowastAutomation and Control Department of Electronic Systems Aalborg University (AAU) 9220Aalborg Denmark (e-mail jcesaaudk jakobesaaudk)

daggerDepartment of Chemical Engineering Norwegian University of Science and Tech-nology (NTNU) 7491 Trondheim Norway (e-mail jorgenbachemengntnunoskogechemengntnuno)

159

160 Trade-off between Energy Consumption and Food Quality Loss

B1 Introduction

Increasing energy costs and consumer awareness on food productssafety and qual-ity aspects impose a big challenge to food industries and especially to supermar-kets which have direct contacts with consumers A well-designed optimal controlscheme continuously maintaining a commercial refrigeration system at its opti-mum operation condition despite changing environmental conditions will achievean important performance improvement both on energy efficiency and food qual-ity reliability

Many efforts on optimization of cooling systems have been focused on optimizingobjective functions such as overall energy consumption system efficiency capac-ity or wear of the individual components seeJakobsen and Rasmussen(1998)Jakobsen et al(2001) Larsen and Thybo(2004) Leducqa et al(2006) Swens-son(1994) They have proved significant improvements of system performanceunder disturbances while there has been little emphasis on the quality aspectoffoodstuffs inside display cabinets

This paper discusses a dynamic optimization of commercial refrigeration systemsfeaturing a balanced system energy consumption and food quality loss A formerdeveloped quality model of food provides a tool for monitoring and controlling thequality loss during the whole process seeCai et al(2006)

The paper is organized as follows Operation and modelling of a refrigerationsystems is presented in SectionB2 In SectionB3 we introduce the problem for-mulation used for optimization Different optimization schemes and results arepresented in SectionB4 Finally some discussions and conclusions follow in Sec-tion B5 and SectionB6

B2 Process description

A simplified sketch of the process is shown in FigureB1 In the evaporator thereis heat exchange between the air inside the display cabinet and the cold refrigerantgiving a slightly super-heated vapor to the compressor After compression the hotvapor is cooled condensed and slightly sub-cooled in the condenser This slightlysub-cooled liquid is then expanded through the expansion valve giving a cold two-phase mixture

The display cabinet is located inside a store and we assume that the store hasaconstant temperature This is relative true for stores with air-conditioningThe

B2 Process description 161

Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13

Display cabinet13Outdoors13

amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13

Figure B1 Sketch of a simplified supermarket refrigeration system studiedin thispaper

condenser and fans are located at the roof of the store Condensationis achievedby heat exchange with ambient air

B21 Degree of freedom analysis

There are 5 degrees of freedom (input) in a general simple refrigeration systemseeJensen and Skogestad(2007) Four of these can be recognized in FigureB1as the compressor speed (NC) condenser fan speed (NCF) evaporator fan speed(NEF) and opening degree of the expansion valve (OD) The fifth one is related tothe active charge in the system

Two of the inputs are already used for control or are otherwise constrained

bull Constant super-heating (∆Tsup = 3C) This is controlled by adjusting theopening degree (OD) of the expansion valve

bull Constant sub-cooling (∆Tsub = 2C) We assume that the condenser is de-signed to give a constant degree of sub-cooling which by design consumesthe degree of freedom related to active charge seeJensen and Skogestad(2007)

So we are left with three unconstrained degrees of freedom that shouldbe used tooptimize the operation These are

1 Compressor speedNC

2 Condenser fan speedNCF

3 Evaporator fan speedNEF

These inputs are controlling three variables

1 Evaporating pressurePE


2 Condensing pressurePC

3 Cabinet temperatureTcabin

However the setpoints for these three variables may be used as manipulatedinputsin our study so the number of degrees of freedom is still three

B22 Mathematical model

The model equations are given in TableB1 please seeLarsen(2005) for the mod-elling of refrigeration systems We assume that the refrigerator has fast dynamicscompared with the display cabinet and food so for the condenser evaporator valveand compressor we have assumed steady-state For the display cabinet and foodwe use a dynamic model as this is where the slow and important (for economics)dynamics will be The food is lumped into one mass and the air inside the cabinettogether with walls are lumped into one mass The main point is that there are twoheat capacities in series For the case with constant display cabinet temperaturewe will also have constant food temperature There are then no dynamics and wemay use steady-state optimization

Some data for the simulations are given in TableB2 please seeLarsen(2005) forfurther data

B23 Influence of setpoints on energy consumption

As stated above this system has three setpoints that may be manipulatedPC PE

andTcabin In FigureB2 surface shows that under 2 different cabinet tempera-tures the variation of energy consumption with varyingPC andPE PointA is theoptimum for cabinet temperatureTcabin1 and pointB is the optimum forTcabin2Tcabin1 is lower thanTcabin2 so the energy consumption is higher in pointA than inpointB

B24 Influence of setpoint on food quality

Food quality decay is determined by its composition factors and many environ-mental factors such as temperature relative humidity light etc Of all the envi-ronmental factors temperature is the most important since it not only stronglyaffects reaction rates but is also directly imposed to the food externally Theotherfactors are at least to some extent controlled by food packaging


Table B1 Model equationsCompressor

WC =mre f middot(his(PePc)minushoe(Pe))

ηis

hic =1minus fq

ηismiddot (his(PePc)minushoe(Pe))+hoe(Pe)

mre f = NC middotVd middotηvol middotρre f (Pe)CondenserWCF = K1CF middot (NCF)3

mairC = K2CF middotNCF

Taoc = Tc +(TambminusTc) middotexp(

minus(αC middot mmCairC)(mairC middotCpair)

)

0 = mre f middot (hic(PePc)minushoc(Pc))minus mairC middotCpair middot (TaocminusTamb)EvaporatorWEF = K1EF middot (NEF)3

mairE = K2EF middotNEF

Taoe= Te+(TcabinminusTe) middotexp(

minus(αE middot mmEairE)(mairE middotCpair)

)

0 = Qeminus mairE middotCpair middot (TcabinminusTaoe)Display cabinetQc2 f = UAc2 f middot (TcabinminusTf ood)Qs2c = UAs2c middot (TstoreminusTcabin)dTf ood

dt = (mCpf ood)minus1 middot Qc2 f

dTcabindt = (mCpcabin)

minus1 middot (minusQc2 f minus QE + Qs2c)

Qf oodloss=int t ft0 100middotDTre f exp(Tf oodminusTre f

z )dt

Table B2 Some data used in the simulationDisplay cabinetlowast

heat transfer areaUAs2c= 160WK-1

heat capacitymCpcabin= 10kJK-1

Foodheat transfer areaUAc2f = 200WKminus1

heat capacitymCpfood = 756kJK-1

quality parameterDTre f = 02day-1quality parameterTre f = 0Cquality parameterZ = 10C


15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2

Figure B2 Energy consumption under different setpoints

Here we focus on the temperature influence to food qualityQfood The only set-point directly influencing food temperature (and thus food quality) isTcabin FigB3 shows the daily quality loss for chilled cod product under 4 casesTfood of2 1C andTsin Tsin1 andTsin2 are the sinusoidal function with mean value of1C amplitude of 1C and 3C respectively period is 24h Note that the qualityloss is higher with higher temperature but there is only minor extra loss over 24hby using a sinusoidal temperature with small amplitude A sinusoidal with largeamplitude has a larger influence on quality due to the non-linearity of the qualityfunction it will not be considered here

0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C

Figure B3 Fresh fish quality loss when stored at different temperatures

B3 Problem formulation

We here consider at a time horizon of three days ambient temperature (Tamb) fol-lows a sinusoidal function with a mean value of 20C period of 24 hours andamplitude of 6C This is a normal temperature profile in Denmark during sum-mer seeDanmarks Meteorologiske Institut(2007)

B3 Problem formulation 165

The objective is to minimize the energy consumption subject to maintaining afixed quality loss by using those 3 DOF This can be formulated mathematicallyas

min(NC(t)NCF(t)NEF(t))

J (B1)

where J =int t f

t0(WC(t)+WCF(t)+WEF(t)︸︷︷︸

Wtot(t)

)dt (B2)

The quality loss of the food could be included in the objective function directlybut we choose to limit it by using constraints The optimization is also subjected toother constraints such as maximum speed of fans and compressor minimum andmaximum value of evaporator and condenser pressure etc

In this paper the food is a fresh cod product Danish food authorities requireit to be kept at a maximum of 2C The control engineer will normally set thetemperature setpoint a little lower for example at 1C

Case 1Traditional operation with constant pressures (PE) (PC) and constanttemperatures (Tcabin= Tfood = 1C)

There are usually large variations in the ambient temperature during the yearso intraditional operation it is necessary to be conservative when choosing the setpointfor condenser pressure To reduce this conservativeness it is common to use onevalue for summer and one for winter We will here assume that the summer settingis used

To get a fair comparison with traditional control which operates at 1C we willillustrate our optimization by considering the following cases

Case 2Tcabin andTfood constant at 1CTwo remaining unconstrained degrees of freedom as functions of time areused for minimizing the energy consumption inB1

Case 3Tfood = 1t fminust0

int t ft0 Tfood(t)dt = 1C

Three remaining unconstrained degrees of freedom as functions of time areused for minimizing the energy consumption inB1

Case 4Qfood loss(t f ) le 755Three remaining unconstrained degrees of freedom as functions of time areused for minimizing the energy consumption inB1 755 is the qualityloss at constant temperature of 1C obtained in cases 1 and 2


B4 Optimization

B41 Optimization

The model is implemented ingPROMSr and the optimization is done by dynamicoptimization (except for Case 1) For the Case 2 we have used piecewiselinearmanipulated variables with a discretisation every hour For the cases with varyingcabinet temperature (Case 3 and 4) we have used sinusoidal functionsu= u0+Amiddotsin(π middot t24+ φ) whereu0 is the nominal inputA is the amplitude of the inputtis the time andφ is the phase shift of the input

Using a sinusoidal function has several advantages

bull There are much fewer variables to optimize on only 3 for each input com-pared with 3 parameters for each time interval for discrete dynamic opti-mization

bull There are no end-effects

In all cases we find that the phase shift is very small

B42 Optimization results

TableB3 compares the four cases in terms of the overall costJ end quality lossmaximum total power (Wtotmax) and maximum compressor power (WCmax) Thetwo latter variables might be important if there are restrictions on the maximumcompressor power or on the total electric power consumption

Some key variables including speed and energy consumption for compressor andfans as well as temperatures are plotted for each case in FigureB5 through FigureB8

Table B3 Traditional operation and optimal operation for three differentcon-straints

Case 1lowast Case 2dagger Case 3Dagger Case 4sect

J [MJ] 2737 2428 2407 2414Qf oodloss(t f ) [] 755 755 761 755WCmax[W] 955 1022 836 879Wtotmax[W] 1233 1136 946 981

For Case 1 (traditional operation) the total energy consumption over threedays is2737MJ Note that the condenser temperature (and pressure) is not changing withtime

B4 Optimization 167

If we keepTcabin = Tfood constant at 1C but allow the pressures (and temper-atures) in the condenser and evaporator to change with time (Case 2) we mayreduce the total energy consumption by 113 to 2428MJ Fig B6 shows thatthe evaporator temperature is constant because we still control the cabinet tem-perature while the condenser temperature varies with ambient temperatureThequality is the same as in Case 1 because of the constant cabinet temperatureThepower variations are larger but nevertheless the maximum total power (Wtotmax)is reduced by 79 to 1136W

Next we also allow the cabinet temperature to vary but add a constraint ontheaverage food temperaturesT food = 10C (Case 3) This reduces the total energyconsumption with another 09 while the food quality loss is slightly higherNote from FigureB7 that the evaporator cabinet and food temperature is varyinga lot

Finally in Case 4 we do not care about the average food temperature but insteadrestrict the quality loss WithQfoodloss(t f )le 755 which is the same end qualitywe obtained for Case 1 we save 118 energy compared with Case 1 but useslightly more than for Case 3 (029) Note from FigureB8 that the amplitudefor food cabinet and evaporator temperature are slightly reduced compared to Case3

An important conclusion is that most of the benefit in terms of energy savingsisobtained by letting the setpoint forPE andPC vary (Case 2) The extra savings bychanging also the cabinet temperatureTcabin (Case 3 and 4) are small Howeverthe peak value for compressor power and total system power is significantly de-creased for Case 3 and 4 This is also very important because a lower compressorcapacity means a lower investment cost and a lower peak value of total powerconsumption will further reduce the bill for supermarket owner according to thefollowing formula

Cop =int year

month(Pel(t) middotEel(t)+max(Pel(t)) middotEeldem(t))dt (B3)

whereCop is the operating costEel is the electricity ratePel is the electric powerEeldemis the electricity demand charge max(Pel(t)) is the maximum electric powerduring one month

B43 Trade-off between energy consumption and food qualityloss

Fig B4 plots the Pareto optimal curve between food quality loss and energyconsumption It shows that reducing quality loss and saving energy is a conflicting


objective to a system An acceptable tradeoff between these two goals can beselected by picking a point somewhere along the line It also shows that Case 1 isfar away from optimization Case 4 is one optimal point while Case 2 and 3 arenear optimal solutions

75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28

Daily Energy Consumption [MJ]

Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2

Pareto Optimal Curve

Figure B4 Optimization between food quality loss and energy consumption

B5 Discussion

Having oscillations in the pressures will impose stress and cause wear on theequip-ment This might not be desirable in many cases but in this study the oscillationsare with a period of one day so this should not be an issue

Experiments on the influence of fluctuating temperatures on food quality werereviewed byUlrich (1981) where marginal reduction in final quality due to fluc-tuations was reported In our case food temperature is only slowly varying andwith an amplitude of less than 1C Thus this will not pose any negative influenceon food quality

B6 Conclusion

We have shown that traditional operation where the pressures are constant givesexcessive energy consumption Allowing for varying pressure in the evaporatorand condenser reduces the total energy consumption by about 11 Varying foodtemperature gives only minor extra improvements in terms of energy consumptionbut the peak value of the total power consumption is reduced with an additional14 for the same food quality loss

Reducing quality loss and saving energy is a conflicting objective Our optimiza-tion result will help the engineer to select an acceptable tradeoff between these two

Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40

(c) High temperatures

Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4

(d) Low temperatures

Figure B5 Traditional operation withTcabin= 1C PE = 24bar andPC = 80bar(Case 1)

goals by picking a point somewhere along the Pareto front line

This paper investigates the potential of finding a balancing point between qualityand energy consumption by open-loop dynamic optimizations It uses the sinusoidambience temperature as one example In real life weather patterns are not exactlya sinusoidal function but real weather conditions can be easily obtainedin advancefrom forecast Practical implementation including selecting controlled variablesand using closed-loop feedback control will be the theme of future research

Bibliography

Cai J Risum J and Thybo C (2006) Quality model of foodstuff in a refrig-erated display cabinet In proc International refrigeration conference PurdueUSA

170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4


Figure B6 Optimal operation forTcabin= 1C (Case 2)

Danmarks Meteorologiske Institut(2007)URL wwwdmidk

Jakobsen A and Rasmussen B (1998) Energy-optimal speed control of fans andcompressors in a refrigeration system Eurotherm 62 Nancy Francepp pp317ndash323

Jakobsen A Rasmussen B Skovrup M J and Fredsted J (2001) Develope-ment of energy optimal capacity control in refrigeration systems In procIn-ternational refrigeration conference Purdue USA


Larsen L F S (2005) Model based control of refrigeration system PhD thesisDepartment of Control Engineering Aalborg University Fredrik Bajers Vej 7CDK 9220 Aalborg Denmark

Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4


Figure B7 Optimal operation forT food = 1C (Case 3)

Larsen L F S and Thybo C (2004) Potential energy saving in refrigerationsystem using optimal setpoint In proc Conference on control applicationsTaipei Taiwan

Leducqa D Guilparta J and Trystramb G (2006) lsquoNon-linear predictive controlof a vapour compression cyclersquoInternational Journal of Refrigeration29 761ndash772

Swensson M C (1994) studies on on-line optimizing control with application toa heat pump PhD thesis Department of refrigeration Engineering NTNU

Ulrich R (1981) lsquoVariations de temperature et qualite des produits surgelesrsquoRe-view Generale due Froid71 371 V389

172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4


Figure B8 Optimal operation forQfood le 755 (Case 4)

Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction


521 Tmin design-method






55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model




76 Nominal optimum maximum production case (Mode II)



78 Discussion





Bibliography


81 Introduction


821 Remark on active charge or ``feed for closed cycles


83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion

A Optimal operation of a mixed fluid cascade LNG plant

A1 Introduction



A31 Manipulated variables (MVs)






A6 Conclusion

Bibliography

B Trade-off between Energy Consumption and Food Quality Loss

B1 Introduction







B4 Optimization

B41 Optimization


B43 Trade-off between energy consumption and food quality loss

B5 Discussion

B6 Conclusion

Bibliography

NTNU

Norwegian University of Science and Technology


Faculty of Natural Sciences and TechnologyDepartment of chemical engineering

ccopy 2008 Joslashrgen Bauck Jensen

ISBN 978-82-471-7801-0 (printed version)ISBN 978-82-471-7815-7 (electronic version)ISSN 1503-8181

Doctoral theses at NTNU 200891

Printed by tapir uttrykk

iii

Abstract








iv



v

Acknowledgement





vi

Contents

List of Figures xi

List of Tables xv








vii

viii Contents















Contents ix
















x Contents







9 Conclusion 147






Contents xi





xii Contents

List of Figures




xiii

xiv List of Figures








List of Figures xv


xvi List of Figures

List of Tables






95



xvii




Chapter 1

Introduction

11 Motivation




1

2 Introduction


12 Thesis overview




method






13 Publication list

Journal papers

Chapter 3


Chapter 4


Chapter 5


4 Introduction

Chapter 8


Conferences

Chapter 4


Chapter 7


Appendix A


Co-authored papers

Appendix B


Chapter 2


21 Introduction







5


QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup



211 Fundamentals




COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively


21 Introduction 7


QH

QC


Cold sink (TC)

Hot source (TH)


∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC



THand

∆SC = QCTC


QH

TH=

QC

TC(23)


W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH








P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)




Tank withsaturation





21 Introduction 9



213 Condenser





∆Tsub

Air fan








214 Evaporator





∆Tsup

Air fan




21 Introduction 11

215 Active charge


TC

Super-heatcontrol


TC

Super-heatcontrol














TC

Super-heatcontrol

Sub-coolingcontrol








[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup


T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant





(a) Flowsheet



P[bar]

h[Jkg-1]









Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2












224 Cascaded cycles


23 Conclusion 17

HX2

HX1



23 Conclusion


18 Bibliography

Bibliography







Chapter 3




19


31 Introduction





QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup


(b)



31 Introduction 21







COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)
























mactive

+mtanks (32)




mactive

+mtanks (33)














z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph








Extra valve removed





QH




QCTC


QC













331 Optimal designs




replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph


QC

QH


z

LC

Pl

Ph












QC

QH

Ws

Super-heatcontrolz

Pl

Ph


QC

QH

Ws

z

LC

Pl

Ph


QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph




optimal





C

TC TsC

QH

QC

z

Ws

Ph

Pl











Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107


Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107





Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120






C] 000 466∆Tsup[






342 Explanation


P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out



∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)




(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)











P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line







35 Discussion 37



min (Ws) (37)



min (Ws) (38)


Amaxi minusAi ge 0


35 Discussion






z

QC

QH

QA

Ws

Pl

Ph












36 Conclusion 39



36 Conclusion




Bibliography




40 Bibliography











Chapter 4




41 Introduction


41


TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl
















are















421 Linear analysis






∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2




spany


L =|Juu|

21

|Gprime|2(41)













431 Modelling















h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1


Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)







z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127
















C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[









Nonlinear analysis



s[k

W]

TH [C]10 15 20 25 30

1

2

3

4




Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH


Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5


Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU


Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5


Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH









ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)





44CO2 case study 53


PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s


44 CO2 case study






441 Modelling


z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1





44CO2 case study 55





RoomTC = TsC = 20C

RoomUAloss= 400WC-1



z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312






Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4






)(45)



44CO2 case study 57



PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime


Linear method



d2 ∆TC = plusmn5C







Non-linear analysis




s[k

W]

TH [C]25 30 35 400

1

2

3

4



TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph



44CO2 case study 59


Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)







z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4


45 Discussion

451 Super-heating


45 Discussion 61


TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl







62 Bibliography


46 Conclusion



Acknowledgment


Bibliography



Bibliography 63















64 Bibliography

Chapter 5





51 Introduction



65




0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)





minu1


)(51)

51 Introduction 67






minu2

(Joperation) (53)






i whereAmaxi is the



minu3

(Joperation) (54)






TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl










bull Qloss= 20kW









minu2

(Ws)








int(U middot∆T)dA


minu3

(Ws)






Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120


Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120









min [C] 100 100 913 267∆Tcon


C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160









minu4

(


Ani

)

(57)










method





vap)) (59)













min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx























55 Discussion

551 ∆Tmin-method






55 Discussion 77






A



Q = UA∆T (510)




78 Bibliography


56 Conclusion





Bibliography





Bibliography 79








80 Bibliography

Chapter 6



61 Introduction





81





LNG Flash gas


Fuel HXNG HX


Fuelcompressor

Condenser Ph

Pl

PmTout


61 Introduction 83

611 Optimal design



subject to cle 0











min


Jflows

+C0 middotsumi

Ani

[$year]

(62)




min(minusmLNG +C0

(A065

HOT +A065NG

))(63)


mfuel le mmaxfuel

cle 0


61 Introduction 85









u =3600min-1

















LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout









bull Pressure drops


















mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120


Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161












































1copy 2copy











(m-2)065



(m-2)065



64 Discussion 95








64 Discussion

641 Compressor













64 Discussion 97

643 Feed pressure



P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100









28MTPA120MW

middot400MW= 93MTPA


65 Conclusion


Bibliography 99

Bibliography










100 Bibliography

Chapter 7



71 Introduction



101






s )







NG30C40bar

Ph

Pl

Pm

N


Main heat exchanger


SW

30C






bull Pressure drops





































73 Model




∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)






f (DNmP1P2T1T2 R) = 0 (72)


73 Model 107


Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)


P1and T2

T1may be plotted




Pr = f (mrNr) (74)

η = f (mrNr) (75)




Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1





2H

2W

Pr0

Pr [-]

mr[-]




H = H0minus12

(

H0 +Pr0

2minus1

)


W = W0(Nr)13 (78)



η = η0

((

1minus

(H minusH0

H0

)2)


)

(79)

73 Model 109




bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-












minu


subject to cle 0




minu

Ws (712)


cle 0















minu

minus mLNG (713)

subject to cle 0













h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107



Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8











Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50


Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30







s [MW] 120 110 130 d1lowast



xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7








s )




















TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N


XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN


XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom


XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup


XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom


mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom




d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN


d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f


d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref


d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref


XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup


XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N




Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30


strategy







N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85



78 Discussion




78 Discussion 121










122 Bibliography

Bibliography














Bibliography 123




Unit Equations





∆Pmiddotρ











124 Bibliography







d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8




81 Introduction


125

































QH

Wsz

Ph

Pl

Evaporator

Condenser















TC

Super-heatcontrol


TC

Super-heatcontrol









z

QC

QH

Ws

Pl

Ph


z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph

Pm














83 Case studies 133

83 Case studies







Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2














83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy


1copy

2copy

3copy

4copy

5copy6copy

7copy



shown in Figure85










Nss = 4



Nss = 4


















83 Case studies 137






Nss = 6


1copy

2copy 3copy

4copy

5copy

6copy NG

LNG
















83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy




























83 Case studies 141
















1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG










84 Discussion 143



+ 6 CompositionNmax











84 Discussion









85 Conclusion




Bibliography 145



Bibliography











146 Bibliography













Chapter 9

Conclusion






147

148 Conclusion




Appendix A



A1 Introduction



149










Nominal conditions







NG

LC

SC

PC1 PC2

LNG


Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW



































bull Pm in SC





Some remarks





Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A


0 50 100 150215220225230235240245250255260

(b) NG1B


Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2



120130140150160170180190200

(d) NG3









NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH













bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm


A6 Conclusion


Bibliography 157



Bibliography






158 Bibliography

Appendix B







159


B1 Introduction









Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13


amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13





























ηis

hic =1minus fq






)





)






z )dt








15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2



0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C







J (B1)

where J =int t f


Wtot(t)

)dt (B2)












B4 Optimization

B41 Optimization













B4 Optimization 167





Cop =int year







75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28


Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2



B5 Discussion



B6 Conclusion



Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4





Bibliography


170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4








Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4







172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4



Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction








55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model







78 Discussion





Bibliography


81 Introduction




83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion


A1 Introduction









A6 Conclusion

Bibliography


B1 Introduction







B4 Optimization

B41 Optimization



B5 Discussion

B6 Conclusion

Bibliography

iii

Abstract








iv



v

Acknowledgement





vi

Contents

List of Figures xi

List of Tables xv








vii

viii Contents















Contents ix
















x Contents







9 Conclusion 147






Contents xi





xii Contents

List of Figures




xiii

xiv List of Figures








List of Figures xv


xvi List of Figures

List of Tables






95



xvii




Chapter 1

Introduction

11 Motivation




1

2 Introduction


12 Thesis overview




method






13 Publication list

Journal papers

Chapter 3


Chapter 4


Chapter 5


4 Introduction

Chapter 8


Conferences

Chapter 4


Chapter 7


Appendix A


Co-authored papers

Appendix B


Chapter 2


21 Introduction







5


QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup



211 Fundamentals




COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively


21 Introduction 7


QH

QC


Cold sink (TC)

Hot source (TH)


∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC



THand

∆SC = QCTC


QH

TH=

QC

TC(23)


W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH








P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)




Tank withsaturation





21 Introduction 9



213 Condenser





∆Tsub

Air fan








214 Evaporator





∆Tsup

Air fan




21 Introduction 11

215 Active charge


TC

Super-heatcontrol


TC

Super-heatcontrol














TC

Super-heatcontrol

Sub-coolingcontrol








[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup


T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant





(a) Flowsheet



P[bar]

h[Jkg-1]









Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2












224 Cascaded cycles


23 Conclusion 17

HX2

HX1



23 Conclusion


18 Bibliography

Bibliography







Chapter 3




19


31 Introduction





QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup


(b)



31 Introduction 21







COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)
























mactive

+mtanks (32)




mactive

+mtanks (33)














z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph








Extra valve removed





QH




QCTC


QC













331 Optimal designs




replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph


QC

QH


z

LC

Pl

Ph












QC

QH

Ws

Super-heatcontrolz

Pl

Ph


QC

QH

Ws

z

LC

Pl

Ph


QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph




optimal





C

TC TsC

QH

QC

z

Ws

Ph

Pl











Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107


Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107





Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120






C] 000 466∆Tsup[






342 Explanation


P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out



∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)




(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)











P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line







35 Discussion 37



min (Ws) (37)



min (Ws) (38)


Amaxi minusAi ge 0


35 Discussion






z

QC

QH

QA

Ws

Pl

Ph












36 Conclusion 39



36 Conclusion




Bibliography




40 Bibliography











Chapter 4




41 Introduction


41


TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl
















are















421 Linear analysis






∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2




spany


L =|Juu|

21

|Gprime|2(41)













431 Modelling















h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1


Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)







z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127
















C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[









Nonlinear analysis



s[k

W]

TH [C]10 15 20 25 30

1

2

3

4




Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH


Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5


Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU


Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5


Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH









ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)





44CO2 case study 53


PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s


44 CO2 case study






441 Modelling


z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1





44CO2 case study 55





RoomTC = TsC = 20C

RoomUAloss= 400WC-1



z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312






Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4






)(45)



44CO2 case study 57



PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime


Linear method



d2 ∆TC = plusmn5C







Non-linear analysis




s[k

W]

TH [C]25 30 35 400

1

2

3

4



TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph



44CO2 case study 59


Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)







z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4


45 Discussion

451 Super-heating


45 Discussion 61


TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl







62 Bibliography


46 Conclusion



Acknowledgment


Bibliography



Bibliography 63















64 Bibliography

Chapter 5





51 Introduction



65




0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)





minu1


)(51)

51 Introduction 67






minu2

(Joperation) (53)






i whereAmaxi is the



minu3

(Joperation) (54)






TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl










bull Qloss= 20kW









minu2

(Ws)








int(U middot∆T)dA


minu3

(Ws)






Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120


Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120









min [C] 100 100 913 267∆Tcon


C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160









minu4

(


Ani

)

(57)










method





vap)) (59)













min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx























55 Discussion

551 ∆Tmin-method






55 Discussion 77






A



Q = UA∆T (510)




78 Bibliography


56 Conclusion





Bibliography





Bibliography 79








80 Bibliography

Chapter 6



61 Introduction





81





LNG Flash gas


Fuel HXNG HX


Fuelcompressor

Condenser Ph

Pl

PmTout


61 Introduction 83

611 Optimal design



subject to cle 0











min


Jflows

+C0 middotsumi

Ani

[$year]

(62)




min(minusmLNG +C0

(A065

HOT +A065NG

))(63)


mfuel le mmaxfuel

cle 0


61 Introduction 85









u =3600min-1

















LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout









bull Pressure drops


















mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120


Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161












































1copy 2copy











(m-2)065



(m-2)065



64 Discussion 95








64 Discussion

641 Compressor













64 Discussion 97

643 Feed pressure



P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100









28MTPA120MW

middot400MW= 93MTPA


65 Conclusion


Bibliography 99

Bibliography










100 Bibliography

Chapter 7



71 Introduction



101






s )







NG30C40bar

Ph

Pl

Pm

N


Main heat exchanger


SW

30C






bull Pressure drops





































73 Model




∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)






f (DNmP1P2T1T2 R) = 0 (72)


73 Model 107


Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)


P1and T2

T1may be plotted




Pr = f (mrNr) (74)

η = f (mrNr) (75)




Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1





2H

2W

Pr0

Pr [-]

mr[-]




H = H0minus12

(

H0 +Pr0

2minus1

)


W = W0(Nr)13 (78)



η = η0

((

1minus

(H minusH0

H0

)2)


)

(79)

73 Model 109




bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-












minu


subject to cle 0




minu

Ws (712)


cle 0















minu

minus mLNG (713)

subject to cle 0













h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107



Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8











Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50


Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30







s [MW] 120 110 130 d1lowast



xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7








s )




















TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N


XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN


XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom


XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup


XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom


mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom




d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN


d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f


d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref


d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref


XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup


XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N




Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30


strategy







N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85



78 Discussion




78 Discussion 121










122 Bibliography

Bibliography














Bibliography 123




Unit Equations





∆Pmiddotρ











124 Bibliography







d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8




81 Introduction


125

































QH

Wsz

Ph

Pl

Evaporator

Condenser















TC

Super-heatcontrol


TC

Super-heatcontrol









z

QC

QH

Ws

Pl

Ph


z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph

Pm














83 Case studies 133

83 Case studies







Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2














83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy


1copy

2copy

3copy

4copy

5copy6copy

7copy



shown in Figure85










Nss = 4



Nss = 4


















83 Case studies 137






Nss = 6


1copy

2copy 3copy

4copy

5copy

6copy NG

LNG
















83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy




























83 Case studies 141
















1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG










84 Discussion 143



+ 6 CompositionNmax











84 Discussion









85 Conclusion




Bibliography 145



Bibliography











146 Bibliography













Chapter 9

Conclusion






147

148 Conclusion




Appendix A



A1 Introduction



149










Nominal conditions







NG

LC

SC

PC1 PC2

LNG


Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW



































bull Pm in SC





Some remarks





Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A


0 50 100 150215220225230235240245250255260

(b) NG1B


Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2



120130140150160170180190200

(d) NG3









NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH













bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm


A6 Conclusion


Bibliography 157



Bibliography






158 Bibliography

Appendix B







159


B1 Introduction









Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13


amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13





























ηis

hic =1minus fq






)





)






z )dt








15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2



0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C







J (B1)

where J =int t f


Wtot(t)

)dt (B2)












B4 Optimization

B41 Optimization













B4 Optimization 167





Cop =int year







75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28


Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2



B5 Discussion



B6 Conclusion



Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4





Bibliography


170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4








Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4







172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4



Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction








55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model







78 Discussion





Bibliography


81 Introduction




83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion


A1 Introduction









A6 Conclusion

Bibliography


B1 Introduction







B4 Optimization

B41 Optimization



B5 Discussion

B6 Conclusion

Bibliography

iv



v

Acknowledgement





vi

Contents

List of Figures xi

List of Tables xv








vii

viii Contents















Contents ix
















x Contents







9 Conclusion 147






Contents xi





xii Contents

List of Figures




xiii

xiv List of Figures








List of Figures xv


xvi List of Figures

List of Tables






95



xvii




Chapter 1

Introduction

11 Motivation




1

2 Introduction


12 Thesis overview




method






13 Publication list

Journal papers

Chapter 3


Chapter 4


Chapter 5


4 Introduction

Chapter 8


Conferences

Chapter 4


Chapter 7


Appendix A


Co-authored papers

Appendix B


Chapter 2


21 Introduction







5


QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup



211 Fundamentals




COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively


21 Introduction 7


QH

QC


Cold sink (TC)

Hot source (TH)


∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC



THand

∆SC = QCTC


QH

TH=

QC

TC(23)


W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH








P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)




Tank withsaturation





21 Introduction 9



213 Condenser





∆Tsub

Air fan








214 Evaporator





∆Tsup

Air fan




21 Introduction 11

215 Active charge


TC

Super-heatcontrol


TC

Super-heatcontrol














TC

Super-heatcontrol

Sub-coolingcontrol








[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup


T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant





(a) Flowsheet



P[bar]

h[Jkg-1]









Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2












224 Cascaded cycles


23 Conclusion 17

HX2

HX1



23 Conclusion


18 Bibliography

Bibliography







Chapter 3




19


31 Introduction





QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup


(b)



31 Introduction 21







COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)
























mactive

+mtanks (32)




mactive

+mtanks (33)














z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph








Extra valve removed





QH




QCTC


QC













331 Optimal designs




replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph


QC

QH


z

LC

Pl

Ph












QC

QH

Ws

Super-heatcontrolz

Pl

Ph


QC

QH

Ws

z

LC

Pl

Ph


QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph




optimal





C

TC TsC

QH

QC

z

Ws

Ph

Pl











Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107


Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107





Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120






C] 000 466∆Tsup[






342 Explanation


P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out



∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)




(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)











P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line







35 Discussion 37



min (Ws) (37)



min (Ws) (38)


Amaxi minusAi ge 0


35 Discussion






z

QC

QH

QA

Ws

Pl

Ph












36 Conclusion 39



36 Conclusion




Bibliography




40 Bibliography











Chapter 4




41 Introduction


41


TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl
















are















421 Linear analysis






∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2




spany


L =|Juu|

21

|Gprime|2(41)













431 Modelling















h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1


Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)







z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127
















C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[









Nonlinear analysis



s[k

W]

TH [C]10 15 20 25 30

1

2

3

4




Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH


Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5


Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU


Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5


Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH









ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)





44CO2 case study 53


PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s


44 CO2 case study






441 Modelling


z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1





44CO2 case study 55





RoomTC = TsC = 20C

RoomUAloss= 400WC-1



z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312






Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4






)(45)



44CO2 case study 57



PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime


Linear method



d2 ∆TC = plusmn5C







Non-linear analysis




s[k

W]

TH [C]25 30 35 400

1

2

3

4



TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph



44CO2 case study 59


Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)







z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4


45 Discussion

451 Super-heating


45 Discussion 61


TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl







62 Bibliography


46 Conclusion



Acknowledgment


Bibliography



Bibliography 63















64 Bibliography

Chapter 5





51 Introduction



65




0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)





minu1


)(51)

51 Introduction 67






minu2

(Joperation) (53)






i whereAmaxi is the



minu3

(Joperation) (54)






TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl










bull Qloss= 20kW









minu2

(Ws)








int(U middot∆T)dA


minu3

(Ws)






Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120


Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120









min [C] 100 100 913 267∆Tcon


C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160









minu4

(


Ani

)

(57)










method





vap)) (59)













min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx























55 Discussion

551 ∆Tmin-method






55 Discussion 77






A



Q = UA∆T (510)




78 Bibliography


56 Conclusion





Bibliography





Bibliography 79








80 Bibliography

Chapter 6



61 Introduction





81





LNG Flash gas


Fuel HXNG HX


Fuelcompressor

Condenser Ph

Pl

PmTout


61 Introduction 83

611 Optimal design



subject to cle 0











min


Jflows

+C0 middotsumi

Ani

[$year]

(62)




min(minusmLNG +C0

(A065

HOT +A065NG

))(63)


mfuel le mmaxfuel

cle 0


61 Introduction 85









u =3600min-1

















LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout









bull Pressure drops


















mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120


Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161












































1copy 2copy











(m-2)065



(m-2)065



64 Discussion 95








64 Discussion

641 Compressor













64 Discussion 97

643 Feed pressure



P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100









28MTPA120MW

middot400MW= 93MTPA


65 Conclusion


Bibliography 99

Bibliography










100 Bibliography

Chapter 7



71 Introduction



101






s )







NG30C40bar

Ph

Pl

Pm

N


Main heat exchanger


SW

30C






bull Pressure drops





































73 Model




∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)






f (DNmP1P2T1T2 R) = 0 (72)


73 Model 107


Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)


P1and T2

T1may be plotted




Pr = f (mrNr) (74)

η = f (mrNr) (75)




Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1





2H

2W

Pr0

Pr [-]

mr[-]




H = H0minus12

(

H0 +Pr0

2minus1

)


W = W0(Nr)13 (78)



η = η0

((

1minus

(H minusH0

H0

)2)


)

(79)

73 Model 109




bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-












minu


subject to cle 0




minu

Ws (712)


cle 0















minu

minus mLNG (713)

subject to cle 0













h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107



Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8











Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50


Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30







s [MW] 120 110 130 d1lowast



xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7








s )




















TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N


XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN


XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom


XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup


XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom


mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom




d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN


d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f


d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref


d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref


XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup


XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N




Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30


strategy







N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85



78 Discussion




78 Discussion 121










122 Bibliography

Bibliography














Bibliography 123




Unit Equations





∆Pmiddotρ











124 Bibliography







d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8




81 Introduction


125

































QH

Wsz

Ph

Pl

Evaporator

Condenser















TC

Super-heatcontrol


TC

Super-heatcontrol









z

QC

QH

Ws

Pl

Ph


z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph

Pm














83 Case studies 133

83 Case studies







Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2














83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy


1copy

2copy

3copy

4copy

5copy6copy

7copy



shown in Figure85










Nss = 4



Nss = 4


















83 Case studies 137






Nss = 6


1copy

2copy 3copy

4copy

5copy

6copy NG

LNG
















83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy




























83 Case studies 141
















1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG










84 Discussion 143



+ 6 CompositionNmax











84 Discussion









85 Conclusion




Bibliography 145



Bibliography











146 Bibliography













Chapter 9

Conclusion






147

148 Conclusion




Appendix A



A1 Introduction



149










Nominal conditions







NG

LC

SC

PC1 PC2

LNG


Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW



































bull Pm in SC





Some remarks





Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A


0 50 100 150215220225230235240245250255260

(b) NG1B


Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2



120130140150160170180190200

(d) NG3









NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH













bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm


A6 Conclusion


Bibliography 157



Bibliography






158 Bibliography

Appendix B







159


B1 Introduction









Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13


amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13





























ηis

hic =1minus fq






)





)






z )dt








15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2



0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C







J (B1)

where J =int t f


Wtot(t)

)dt (B2)












B4 Optimization

B41 Optimization













B4 Optimization 167





Cop =int year







75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28


Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2



B5 Discussion



B6 Conclusion



Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4





Bibliography


170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4








Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4







172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4



Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction








55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model







78 Discussion





Bibliography


81 Introduction




83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion


A1 Introduction









A6 Conclusion

Bibliography


B1 Introduction







B4 Optimization

B41 Optimization



B5 Discussion

B6 Conclusion

Bibliography

v

Acknowledgement





vi

Contents

List of Figures xi

List of Tables xv








vii

viii Contents















Contents ix
















x Contents







9 Conclusion 147






Contents xi





xii Contents

List of Figures




xiii

xiv List of Figures








List of Figures xv


xvi List of Figures

List of Tables






95



xvii




Chapter 1

Introduction

11 Motivation




1

2 Introduction


12 Thesis overview




method






13 Publication list

Journal papers

Chapter 3


Chapter 4


Chapter 5


4 Introduction

Chapter 8


Conferences

Chapter 4


Chapter 7


Appendix A


Co-authored papers

Appendix B


Chapter 2


21 Introduction







5


QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup



211 Fundamentals




COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively


21 Introduction 7


QH

QC


Cold sink (TC)

Hot source (TH)


∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC



THand

∆SC = QCTC


QH

TH=

QC

TC(23)


W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH








P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)




Tank withsaturation





21 Introduction 9



213 Condenser





∆Tsub

Air fan








214 Evaporator





∆Tsup

Air fan




21 Introduction 11

215 Active charge


TC

Super-heatcontrol


TC

Super-heatcontrol














TC

Super-heatcontrol

Sub-coolingcontrol








[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup


T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant





(a) Flowsheet



P[bar]

h[Jkg-1]









Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2












224 Cascaded cycles


23 Conclusion 17

HX2

HX1



23 Conclusion


18 Bibliography

Bibliography







Chapter 3




19


31 Introduction





QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup


(b)



31 Introduction 21







COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)
























mactive

+mtanks (32)




mactive

+mtanks (33)














z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph








Extra valve removed





QH




QCTC


QC













331 Optimal designs




replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph


QC

QH


z

LC

Pl

Ph












QC

QH

Ws

Super-heatcontrolz

Pl

Ph


QC

QH

Ws

z

LC

Pl

Ph


QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph




optimal





C

TC TsC

QH

QC

z

Ws

Ph

Pl











Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107


Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107





Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120






C] 000 466∆Tsup[






342 Explanation


P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out



∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)




(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)











P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line







35 Discussion 37



min (Ws) (37)



min (Ws) (38)


Amaxi minusAi ge 0


35 Discussion






z

QC

QH

QA

Ws

Pl

Ph












36 Conclusion 39



36 Conclusion




Bibliography




40 Bibliography











Chapter 4




41 Introduction


41


TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl
















are















421 Linear analysis






∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2




spany


L =|Juu|

21

|Gprime|2(41)













431 Modelling















h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1


Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)







z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127
















C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[









Nonlinear analysis



s[k

W]

TH [C]10 15 20 25 30

1

2

3

4




Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH


Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5


Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU


Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5


Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH









ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)





44CO2 case study 53


PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s


44 CO2 case study






441 Modelling


z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1





44CO2 case study 55





RoomTC = TsC = 20C

RoomUAloss= 400WC-1



z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312






Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4






)(45)



44CO2 case study 57



PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime


Linear method



d2 ∆TC = plusmn5C







Non-linear analysis




s[k

W]

TH [C]25 30 35 400

1

2

3

4



TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph



44CO2 case study 59


Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)







z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4


45 Discussion

451 Super-heating


45 Discussion 61


TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl







62 Bibliography


46 Conclusion



Acknowledgment


Bibliography



Bibliography 63















64 Bibliography

Chapter 5





51 Introduction



65




0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)





minu1


)(51)

51 Introduction 67






minu2

(Joperation) (53)






i whereAmaxi is the



minu3

(Joperation) (54)






TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl










bull Qloss= 20kW









minu2

(Ws)








int(U middot∆T)dA


minu3

(Ws)






Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120


Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120









min [C] 100 100 913 267∆Tcon


C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160









minu4

(


Ani

)

(57)










method





vap)) (59)













min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx























55 Discussion

551 ∆Tmin-method






55 Discussion 77






A



Q = UA∆T (510)




78 Bibliography


56 Conclusion





Bibliography





Bibliography 79








80 Bibliography

Chapter 6



61 Introduction





81





LNG Flash gas


Fuel HXNG HX


Fuelcompressor

Condenser Ph

Pl

PmTout


61 Introduction 83

611 Optimal design



subject to cle 0











min


Jflows

+C0 middotsumi

Ani

[$year]

(62)




min(minusmLNG +C0

(A065

HOT +A065NG

))(63)


mfuel le mmaxfuel

cle 0


61 Introduction 85









u =3600min-1

















LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout









bull Pressure drops


















mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120


Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161












































1copy 2copy











(m-2)065



(m-2)065



64 Discussion 95








64 Discussion

641 Compressor













64 Discussion 97

643 Feed pressure



P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100









28MTPA120MW

middot400MW= 93MTPA


65 Conclusion


Bibliography 99

Bibliography










100 Bibliography

Chapter 7



71 Introduction



101






s )







NG30C40bar

Ph

Pl

Pm

N


Main heat exchanger


SW

30C






bull Pressure drops





































73 Model




∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)






f (DNmP1P2T1T2 R) = 0 (72)


73 Model 107


Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)


P1and T2

T1may be plotted




Pr = f (mrNr) (74)

η = f (mrNr) (75)




Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1





2H

2W

Pr0

Pr [-]

mr[-]




H = H0minus12

(

H0 +Pr0

2minus1

)


W = W0(Nr)13 (78)



η = η0

((

1minus

(H minusH0

H0

)2)


)

(79)

73 Model 109




bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-












minu


subject to cle 0




minu

Ws (712)


cle 0















minu

minus mLNG (713)

subject to cle 0













h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107



Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8











Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50


Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30







s [MW] 120 110 130 d1lowast



xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7








s )




















TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N


XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN


XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom


XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup


XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom


mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom




d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN


d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f


d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref


d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref


XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup


XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N




Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30


strategy







N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85



78 Discussion




78 Discussion 121










122 Bibliography

Bibliography














Bibliography 123




Unit Equations





∆Pmiddotρ











124 Bibliography







d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8




81 Introduction


125

































QH

Wsz

Ph

Pl

Evaporator

Condenser















TC

Super-heatcontrol


TC

Super-heatcontrol









z

QC

QH

Ws

Pl

Ph


z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph

Pm














83 Case studies 133

83 Case studies







Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2














83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy


1copy

2copy

3copy

4copy

5copy6copy

7copy



shown in Figure85










Nss = 4



Nss = 4


















83 Case studies 137






Nss = 6


1copy

2copy 3copy

4copy

5copy

6copy NG

LNG
















83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy




























83 Case studies 141
















1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG










84 Discussion 143



+ 6 CompositionNmax











84 Discussion









85 Conclusion




Bibliography 145



Bibliography











146 Bibliography













Chapter 9

Conclusion






147

148 Conclusion




Appendix A



A1 Introduction



149










Nominal conditions







NG

LC

SC

PC1 PC2

LNG


Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW



































bull Pm in SC





Some remarks





Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A


0 50 100 150215220225230235240245250255260

(b) NG1B


Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2



120130140150160170180190200

(d) NG3









NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH













bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm


A6 Conclusion


Bibliography 157



Bibliography






158 Bibliography

Appendix B







159


B1 Introduction









Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13


amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13





























ηis

hic =1minus fq






)





)






z )dt








15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2



0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C







J (B1)

where J =int t f


Wtot(t)

)dt (B2)












B4 Optimization

B41 Optimization













B4 Optimization 167





Cop =int year







75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28


Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2



B5 Discussion



B6 Conclusion



Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4





Bibliography


170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4








Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4







172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4



Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction








55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model







78 Discussion





Bibliography


81 Introduction




83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion


A1 Introduction









A6 Conclusion

Bibliography


B1 Introduction







B4 Optimization

B41 Optimization



B5 Discussion

B6 Conclusion

Bibliography

vi

Contents

List of Figures xi

List of Tables xv








vii

viii Contents















Contents ix
















x Contents







9 Conclusion 147






Contents xi





xii Contents

List of Figures




xiii

xiv List of Figures








List of Figures xv


xvi List of Figures

List of Tables






95



xvii




Chapter 1

Introduction

11 Motivation




1

2 Introduction


12 Thesis overview




method






13 Publication list

Journal papers

Chapter 3


Chapter 4


Chapter 5


4 Introduction

Chapter 8


Conferences

Chapter 4


Chapter 7


Appendix A


Co-authored papers

Appendix B


Chapter 2


21 Introduction







5


QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup



211 Fundamentals




COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively


21 Introduction 7


QH

QC


Cold sink (TC)

Hot source (TH)


∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC



THand

∆SC = QCTC


QH

TH=

QC

TC(23)


W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH








P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)




Tank withsaturation





21 Introduction 9



213 Condenser





∆Tsub

Air fan








214 Evaporator





∆Tsup

Air fan




21 Introduction 11

215 Active charge


TC

Super-heatcontrol


TC

Super-heatcontrol














TC

Super-heatcontrol

Sub-coolingcontrol








[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup


T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant





(a) Flowsheet



P[bar]

h[Jkg-1]









Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2












224 Cascaded cycles


23 Conclusion 17

HX2

HX1



23 Conclusion


18 Bibliography

Bibliography







Chapter 3




19


31 Introduction





QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup


(b)



31 Introduction 21







COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)
























mactive

+mtanks (32)




mactive

+mtanks (33)














z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph








Extra valve removed





QH




QCTC


QC













331 Optimal designs




replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph


QC

QH


z

LC

Pl

Ph












QC

QH

Ws

Super-heatcontrolz

Pl

Ph


QC

QH

Ws

z

LC

Pl

Ph


QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph




optimal





C

TC TsC

QH

QC

z

Ws

Ph

Pl











Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107


Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107





Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120






C] 000 466∆Tsup[






342 Explanation


P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out



∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)




(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)











P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line







35 Discussion 37



min (Ws) (37)



min (Ws) (38)


Amaxi minusAi ge 0


35 Discussion






z

QC

QH

QA

Ws

Pl

Ph












36 Conclusion 39



36 Conclusion




Bibliography




40 Bibliography











Chapter 4




41 Introduction


41


TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl
















are















421 Linear analysis






∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2




spany


L =|Juu|

21

|Gprime|2(41)













431 Modelling















h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1


Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)







z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127
















C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[









Nonlinear analysis



s[k

W]

TH [C]10 15 20 25 30

1

2

3

4




Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH


Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5


Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU


Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5


Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH









ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)





44CO2 case study 53


PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s


44 CO2 case study






441 Modelling


z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1





44CO2 case study 55





RoomTC = TsC = 20C

RoomUAloss= 400WC-1



z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312






Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4






)(45)



44CO2 case study 57



PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime


Linear method



d2 ∆TC = plusmn5C







Non-linear analysis




s[k

W]

TH [C]25 30 35 400

1

2

3

4



TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph



44CO2 case study 59


Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)







z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4


45 Discussion

451 Super-heating


45 Discussion 61


TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl







62 Bibliography


46 Conclusion



Acknowledgment


Bibliography



Bibliography 63















64 Bibliography

Chapter 5





51 Introduction



65




0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)





minu1


)(51)

51 Introduction 67






minu2

(Joperation) (53)






i whereAmaxi is the



minu3

(Joperation) (54)






TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl










bull Qloss= 20kW









minu2

(Ws)








int(U middot∆T)dA


minu3

(Ws)






Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120


Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120









min [C] 100 100 913 267∆Tcon


C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160









minu4

(


Ani

)

(57)










method





vap)) (59)













min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx























55 Discussion

551 ∆Tmin-method






55 Discussion 77






A



Q = UA∆T (510)




78 Bibliography


56 Conclusion





Bibliography





Bibliography 79








80 Bibliography

Chapter 6



61 Introduction





81





LNG Flash gas


Fuel HXNG HX


Fuelcompressor

Condenser Ph

Pl

PmTout


61 Introduction 83

611 Optimal design



subject to cle 0











min


Jflows

+C0 middotsumi

Ani

[$year]

(62)




min(minusmLNG +C0

(A065

HOT +A065NG

))(63)


mfuel le mmaxfuel

cle 0


61 Introduction 85









u =3600min-1

















LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout









bull Pressure drops


















mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120


Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161












































1copy 2copy











(m-2)065



(m-2)065



64 Discussion 95








64 Discussion

641 Compressor













64 Discussion 97

643 Feed pressure



P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100









28MTPA120MW

middot400MW= 93MTPA


65 Conclusion


Bibliography 99

Bibliography










100 Bibliography

Chapter 7



71 Introduction



101






s )







NG30C40bar

Ph

Pl

Pm

N


Main heat exchanger


SW

30C






bull Pressure drops





































73 Model




∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)






f (DNmP1P2T1T2 R) = 0 (72)


73 Model 107


Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)


P1and T2

T1may be plotted




Pr = f (mrNr) (74)

η = f (mrNr) (75)




Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1





2H

2W

Pr0

Pr [-]

mr[-]




H = H0minus12

(

H0 +Pr0

2minus1

)


W = W0(Nr)13 (78)



η = η0

((

1minus

(H minusH0

H0

)2)


)

(79)

73 Model 109




bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-












minu


subject to cle 0




minu

Ws (712)


cle 0















minu

minus mLNG (713)

subject to cle 0













h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107



Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8











Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50


Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30







s [MW] 120 110 130 d1lowast



xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7








s )




















TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N


XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN


XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom


XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup


XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom


mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom




d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN


d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f


d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref


d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref


XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup


XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N




Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30


strategy







N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85



78 Discussion




78 Discussion 121










122 Bibliography

Bibliography














Bibliography 123




Unit Equations





∆Pmiddotρ











124 Bibliography







d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8




81 Introduction


125

































QH

Wsz

Ph

Pl

Evaporator

Condenser















TC

Super-heatcontrol


TC

Super-heatcontrol









z

QC

QH

Ws

Pl

Ph


z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph

Pm














83 Case studies 133

83 Case studies







Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2














83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy


1copy

2copy

3copy

4copy

5copy6copy

7copy



shown in Figure85










Nss = 4



Nss = 4


















83 Case studies 137






Nss = 6


1copy

2copy 3copy

4copy

5copy

6copy NG

LNG
















83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy




























83 Case studies 141
















1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG










84 Discussion 143



+ 6 CompositionNmax











84 Discussion









85 Conclusion




Bibliography 145



Bibliography











146 Bibliography













Chapter 9

Conclusion






147

148 Conclusion




Appendix A



A1 Introduction



149










Nominal conditions







NG

LC

SC

PC1 PC2

LNG


Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW



































bull Pm in SC





Some remarks





Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A


0 50 100 150215220225230235240245250255260

(b) NG1B


Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2



120130140150160170180190200

(d) NG3









NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH













bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm


A6 Conclusion


Bibliography 157



Bibliography






158 Bibliography

Appendix B







159


B1 Introduction









Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13


amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13





























ηis

hic =1minus fq






)





)






z )dt








15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2



0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C







J (B1)

where J =int t f


Wtot(t)

)dt (B2)












B4 Optimization

B41 Optimization













B4 Optimization 167





Cop =int year







75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28


Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2



B5 Discussion



B6 Conclusion



Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4





Bibliography


170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4








Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4







172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4



Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction








55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model







78 Discussion





Bibliography


81 Introduction




83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion


A1 Introduction









A6 Conclusion

Bibliography


B1 Introduction







B4 Optimization

B41 Optimization



B5 Discussion

B6 Conclusion

Bibliography

Contents

List of Figures xi

List of Tables xv








vii

viii Contents















Contents ix
















x Contents







9 Conclusion 147






Contents xi





xii Contents

List of Figures




xiii

xiv List of Figures








List of Figures xv


xvi List of Figures

List of Tables






95



xvii




Chapter 1

Introduction

11 Motivation




1

2 Introduction


12 Thesis overview




method






13 Publication list

Journal papers

Chapter 3


Chapter 4


Chapter 5


4 Introduction

Chapter 8


Conferences

Chapter 4


Chapter 7


Appendix A


Co-authored papers

Appendix B


Chapter 2


21 Introduction







5


QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a) Flowsheet

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup



211 Fundamentals




COPh =h1minush2

h1minush4=

m(h1minush2)

m(h1minush4)=

QH

Ws(21)

and COPc =h4minush3

h1minush4=

m(h4minush3)

m(h1minush4)=

QC

Ws(22)

respectively


21 Introduction 7


QH

QC


Cold sink (TC)

Hot source (TH)


∆SH = minusQHTH

∆Smachine= 0

∆SC = QCTC



THand

∆SC = QCTC


QH

TH=

QC

TC(23)


W =

(

1minusTC

TH

)

QH (24)

Here 1minus TCTH








P[bar]

h[Jkg-1]

Phase envelope

(ii) (iii) (i)




Tank withsaturation





21 Introduction 9



213 Condenser





∆Tsub

Air fan








214 Evaporator





∆Tsup

Air fan




21 Introduction 11

215 Active charge


TC

Super-heatcontrol


TC

Super-heatcontrol














TC

Super-heatcontrol

Sub-coolingcontrol








[C

]

Position[-]

Cooling load (TC)

Pure refrigerant

∆Tsup


T[

C]

Position[-]

Cooling load (TC)

Pure refrigerant

Mixed refrigerant





(a) Flowsheet



P[bar]

h[Jkg-1]









Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

(a) Flowsheet

T[

C]

Position[-]

Cooling load

Evaporator 1

Evaporator 2












224 Cascaded cycles


23 Conclusion 17

HX2

HX1



23 Conclusion


18 Bibliography

Bibliography







Chapter 3




19


31 Introduction





QC

QH

12

3 4

Wsz

Ph

Pl

Evaporator

Condenser

Chokevalve

Compressor

(a)

12

3 4

P

h[Jkg-1]

Ph

Pl

∆Tsub

∆Tsup


(b)



31 Introduction 21







COP=Qc

Ws=

m(h4minush3)

m(h1minush4)(31)
























mactive

+mtanks (32)




mactive

+mtanks (33)














z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph








Extra valve removed





QH




QCTC


QC













331 Optimal designs




replacements

QC

QH

Ws

Sub-coolingcontrol

zPl

Ph


QC

QH


z

LC

Pl

Ph












QC

QH

Ws

Super-heatcontrolz

Pl

Ph


QC

QH

Ws

z

LC

Pl

Ph


QC

QH

Ws

Super-heatcontrol

Sub-coolingcontrol

zPl

Ph




optimal





C

TC TsC

QH

QC

z

Ws

Ph

Pl











Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107


Enthalpy[Jmol-1]

Pre

ssur

e[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107





Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120


Position[-]

Tem

pera

ture[

C]

0 02 04 06 08 130405060708090

100110120






C] 000 466∆Tsup[






342 Explanation


P

h[Jkg-1]

∆ws∆qC qC ws

TC-line

TH-line

Con out



∆COP=qC +∆qC

ws+∆wsminus

qC

wsgt 0 (34)




(partqC

part∆Tsub

)

UAgt COPmiddot

(partws

part∆Tsub

)

UA(36)











P

h[Jkg-1]

∆wsws∆qC qC

TC-line

TH-line







35 Discussion 37



min (Ws) (37)



min (Ws) (38)


Amaxi minusAi ge 0


35 Discussion






z

QC

QH

QA

Ws

Pl

Ph












36 Conclusion 39



36 Conclusion




Bibliography




40 Bibliography











Chapter 4




41 Introduction


41


TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

z

Ws

QC

Qloss

Ph

Pl

(chokevalve)

(compressor)

(evaporator)

(condenser)

Vl
















are















421 Linear analysis






∆yopt =

radic

sumi

(part∆yopt

partdimiddot∆di

)2




spany


L =|Juu|

21

|Gprime|2(41)













431 Modelling















h [Jmol-1]

P[P

a]

-7 -65 -6 -55 -5 -45times104

105

106

107

2

3 4

1


Position[-]

T[

C]

0 02 04 06 08 12030405060708090

100110120

T1

T2

Tsat(Ph)







z[-] 0372Ph [bar] 1070Pl [bar] 235

QH [kW] 1796m[kgs-1] 00127
















C] 000 000 000 000 000 000 100 100 000T1 [C] 1026 -14374 38 173 62 422 100 432 333Ph [bar] 1071 -1739 412 041 0460 417 100 517 337z[-] 0372 1 00517 00429 00632 0092 005 0142 703T2 [C] 209 28795 104 020 0300 104 100 114 253Vl [m

3] 100 51455 9e-03 0011 12e-03 00143 005 0064 801∆Tsub[









Nonlinear analysis



s[k

W]

TH [C]10 15 20 25 30

1

2

3

4




Loss

[kW

]

TH [C]10 15 20 25 30

01

02

03

04

05

Ph

∆Tsub

z

zNo sub-cooling

Vl

Vconl

T2minusTH


Ws[k

W]

TC [C]-15 -10 -5

1

2

3

4

5


Loss

[kW

]

TC [C]-15 -10 -5

01

02

03

04

05

Phz

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH

AAAAAU


Ws[k

W]

UA[W C-1]400 500 600

1

2

3

4

5


Loss

[kW

]

UA[W C-1]400 500 600

01

02

03

04

05

Ph z

z

No sub-cooling

∆Tsub

Vl

Vconl

T2minusTH









ss[k

W]

Vl [m3]095 1 105

01

02

03

04

05

y = Vl

(a)

Loss

[kW

]

Vconl [m3]062 064 066 068 07 072

01

02

03

04

05

y = Vl con

(b)

Loss

[kW

]

T2minusTH [C]-15 -1 -05 0

01

02

03

04

05

y = T2minusTH

(c)

Loss

[kW

]

∆Tsub[C]

5 6 7

01

02

03

04

05

y = ∆Tsub

(d)





44CO2 case study 53


PSfrag

TC

TC

PC

TC (building)

minusT1

T2

T3 T4

TsC

QH

z

Ws

QC

Qloss

Ph

Pl

Psh

TH (ambient)

(T2minusTH)s


44 CO2 case study






441 Modelling


z

TC

TH

TH

Qihx

Qloss

Ws

QC

QHPh

Pl

TC

T1T2

T3 T4

TsC

Vl

T prime2

T prime4

(Gascooler)

(Internalhex)

(a) TheCO2 cycle

h [Jmol-1]

Pre

ssur

e[P

a]

0C

-41 -405 -4 -395times105

106

107 2

3 4

1





44CO2 case study 55





RoomTC = TsC = 20C

RoomUAloss= 400WC-1



z[-] 034Ph [bar] 9761Pl [bar] 5083QH [W] -4958

Qihx [W] 889m[kgs-1] 0025

T1 [C] 896T2 [C] 255T3 [C] 150T4 [C] 312






Positionφ [-]

T[

C]

0 02 04 06 08 130

40

50

60

70

80

90T1

T prime2

(a) Gas cooler

Positionφ [-]

T[

C]

0 02 04 06 08 110

15

20

25

30

35

40

T2

T prime2

T4

T prime4






)(45)



44CO2 case study 57



PhT prime2 [barC-1] 032 -0291 0140 -0047 0093 0174 00033 0177 025

Ph [bar] 9761 -7885 483 -155 310 594 10 604 131T prime

2 [C] 355 367 1627 -293 764 1821 1 192 191T prime


Linear method



d2 ∆TC = plusmn5C







Non-linear analysis




s[k

W]

TH [C]25 30 35 400

1

2

3

4



TH [C]

Loss

[kW

]

25 30 35 400

01

02

03

04

05Ph

Ph

VlT2

z

z mgco


Ws[k

W]

TC [C]15 20 250

1

2

3

4

5


TC [C]

Loss

[kW

]

15 20 250

01

02

03

04

05Ph

Vl


Ws[k

W]

UA[W C-1]240 280 320 360 400 4400

1

2

3

4

5


UA[W C-1]

Loss

[kW

]

240 280 320 360 400 4400

01

02

03

04

05

Ph



44CO2 case study 59


Vl [m3]

Loss

[kW

]

y = Vl

0070075 008

01

02

03

04

05

(a)

z[-]

Loss

[kW

]

y = z

031 033 035 037

01

02

03

04

05

(b)

Phcombine[bar]

Loss

[kW

]

c = Phcombine

90 95 100 105

01

02

03

04

05

(c)

mgco[kg]

Loss

[kW

]

y = mgco

45 5

01

02

03

04

05

(d)







z

TC

TH

TH

Qloss

Ws

QC

QH

Pl

TC

T1

T3 T4

TsC

k

PC

Ph

T2

Pshcombine

Vl

T prime4


45 Discussion

451 Super-heating


45 Discussion 61


TC

TCTC (building)

minus

T1T2

T3 T4

TsC

QH

z

Ws

QCQloss

Ph

Pl







62 Bibliography


46 Conclusion



Acknowledgment


Bibliography



Bibliography 63















64 Bibliography

Chapter 5





51 Introduction



65




0 1 Position[-]

T[

C]

T1 = constant

T2(Small area)

T2(Large area)

∆Tmin(Large area)

∆Tmin(Small area)





minu1


)(51)

51 Introduction 67






minu2

(Joperation) (53)






i whereAmaxi is the



minu3

(Joperation) (54)






TCTC (building)

TH (ambient)

TH (ambient)

T1T2

T3

T4Ts

C

QH

(condenser)

z(chokevalve)

(compressor)Ws

QC (evaporator)

Qloss

Ph

Pl










bull Qloss= 20kW









minu2

(Ws)








int(U middot∆T)dA


minu3

(Ws)






Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120


Positionφ [-]

T[

C]

0 02 04 06 08 130405060708090

100110120









min [C] 100 100 913 267∆Tcon


C] 00 89 96 285m[mols-1] 107 103 103 110Ws[W] 6019 5824 5803 12479COP[-] 332 343 345 160









minu4

(


Ani

)

(57)










method





vap)) (59)













min [C] 500 356 241 207 500∆Tvap

min [C] 500 500 578 501 115∆T ihx























55 Discussion

551 ∆Tmin-method






55 Discussion 77






A



Q = UA∆T (510)




78 Bibliography


56 Conclusion





Bibliography





Bibliography 79








80 Bibliography

Chapter 6



61 Introduction





81





LNG Flash gas


Fuel HXNG HX


Fuelcompressor

Condenser Ph

Pl

PmTout


61 Introduction 83

611 Optimal design



subject to cle 0











min


Jflows

+C0 middotsumi

Ani

[$year]

(62)




min(minusmLNG +C0

(A065

HOT +A065NG

))(63)


mfuel le mmaxfuel

cle 0


61 Introduction 85









u =3600min-1

















LNG Flash gas

Pre-treated NG SW

NG HX


Condenser

Ph

Pl

PmTout









bull Pressure drops


















mREF [kgs-1] 478 475 472 443 251 298 320 611 617Tout[

C] -144 -156 -156 -156 -157 -157 -157 -157 -156∆Tsup[

C] 100 100 116 257 100 100 100 100 100∆Tmin [C] 196 197 195 203 197 194 194 200 204

η [] 828 828 828 828 828 828 828 828 828Ws[MW] 775 775 775 775 775 775 775 120 120


Pl [bar] 40 40 40 40 224 181 317 411 416Head[kJkg-1] 134 135 136 145 256 216 200 162 161












































1copy 2copy











(m-2)065



(m-2)065



64 Discussion 95








64 Discussion

641 Compressor













64 Discussion 97

643 Feed pressure



P[bar]

mLN

G[k

gs-1

]

40 45 50 55 6075

80

85

90

95

100









28MTPA120MW

middot400MW= 93MTPA


65 Conclusion


Bibliography 99

Bibliography










100 Bibliography

Chapter 7



71 Introduction



101






s )







NG30C40bar

Ph

Pl

Pm

N


Main heat exchanger


SW

30C






bull Pressure drops





































73 Model




∆Psuc= ∆Psuc0

(Vsuc

Vsuc0

)2

(71)






f (DNmP1P2T1T2 R) = 0 (72)


73 Model 107


Pr =P2

P1 Tr =

T2

T1 mr =

mradic

RT1

D2P1and Nr =

NDradic

RT1

(73)


P1and T2

T1may be plotted




Pr = f (mrNr) (74)

η = f (mrNr) (75)




Pr = Pr0 +H

(

1+32

(mrW

minus1

)

minus12

(mrW

minus1

)3)

(76)

wherePr = P2P1





2H

2W

Pr0

Pr [-]

mr[-]




H = H0minus12

(

H0 +Pr0

2minus1

)


W = W0(Nr)13 (78)



η = η0

((

1minus

(H minusH0

H0

)2)


)

(79)

73 Model 109




bull H0 = 18125

bull W0 = 00698

bull η0 = 822

Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8

10

N = 100

1

10 N = 100-












minu


subject to cle 0




minu

Ws (712)


cle 0















minu

minus mLNG (713)

subject to cle 0













h[Jmol-1]

P[P

a]

-11 -105 -10 -95 -9 -85 -8 -75times104

105

106

107



Pr[

-]

mr[-]

η[-]

mr[-]0 002 004 006 008 01 012 014 016

0 002 004 006 008 01 012 014 016

0

05

1

0

2

4

6

8











Position[-]

T[

C]

0 20 40 60 80 100-200

-150

-100

-50

0

50


Position[-]

T[

C]

0 20 40 60 80 1000

5

10

15

20

25

30







s [MW] 120 110 130 d1lowast



xCH4 [] 327 (319) 294 (287) 360 (351)d4

xC2H6 [] 343 (352) 309 (317) 377 (387)d5

xC4H10 [] 233 (247) 210 (222) 256 (272)d6

xN2 [] 97 (82) 87 (74) 107 (90) d7








s )




















TC [C]

Ws[M

W]

25 30 35100

105

110

115

120

Pl

mref

AAAU

N

Toutcom

∆Tsup

N


XCH4 []

Ws[M

W]

29 30 31 32 33 34 35100

105

110

115

120

∆Tsup

Toutcom

RNN


XC2H6 []

Ws[M

W]

32 33 34 35 36 37 38100

105

110

115

120

N

N

∆Tsup

Toutcom


XC4H10 []

Ws[M

W]

2324 25 26 27100

105

110

115

120

NN

Toutcom

∆Tsup


XN2 []

Ws[M

W]

75 8 85 9100

105

110

115

120

N Nmref

R

∆Tsup

Toutcom


mLNG [kgs-1]

Ws[M

W]

67 68 69 70 71 72 73100

105

110

115

120

N

N

mref

Pl

∆Tsup

Toutcom




d1 = Ws[MW]

LNG

[kg

s-1]

110 115 120 125 13065

70

75

80

85

Pl

Toutcom

∆Tsup

mre fN


d2 = Tin [C]

LNG

[kg

s-1]

25 30 3565

70

75

80

85

Pl

N∆Tsup

Toutcom

mre f


d4 = xCH4 []

LNG

[kg

s-1]

30 31 32 33 34 35 3665

70

75

80

85

N

PlToutcom

∆Tsup mref


d5 = xC2H6 []

LNG

[kg

s-1]

31 32 33 34 35 36 3765

70

75

80

85NPl

Toutcom

R∆Tsup

mref


XC4H10 []

LNG

[kg

s-1]

2122 23 24 2565

70

75

80

85

Pl

Nmref Tout

com

∆Tsup


XN2 []

LNG

[kg

s-1]

9 95 10 10565

70

75

80

85

Toutcom

mref

Pl

N ∆Tsup

N




Ws[MW]

∆Tsu

p[C

]

110 115 120 125 1300

5

10

15

20

25

30


strategy







N []

LNG

[kg

s-1]

80 85 90 95100105110 115 12065

70

75

80

85



78 Discussion




78 Discussion 121










122 Bibliography

Bibliography














Bibliography 123




Unit Equations





∆Pmiddotρ











124 Bibliography







d3 = Tin d7 = xN2

d4 = xCH4 d8 = mLNG

Chapter 8




81 Introduction


125

































QH

Wsz

Ph

Pl

Evaporator

Condenser















TC

Super-heatcontrol


TC

Super-heatcontrol









z

QC

QH

Ws

Pl

Ph


z

QC

QH

Ws

Pl

Ph

Pm


z

QC

QH

Ws

Pl

Ph

Pm














83 Case studies 133

83 Case studies







Evaporator 1

Evaporator 2

1copy2copy

3copy 4copy

5copy

6copy

Ph

Pl

T

Processin

Processout

(a) FlowsheetT

[C

]

Position[-]

Process stream

Evaporator 1

Evaporator 2














83 Case studies 135

1copy

2copy

3copy

4copy

5copy6copy

7copy


1copy

2copy

3copy

4copy

5copy6copy

7copy



shown in Figure85










Nss = 4



Nss = 4


















83 Case studies 137






Nss = 6


1copy

2copy 3copy

4copy

5copy

6copy NG

LNG
















83 Case studies 139

1copy

2copy 3copy 4copy

5copy 6copy 7copy

8copy

9copy

10copy

11copy

12copy13copy




























83 Case studies 141
















1copy2copy

3copy

4copy

5copy

6copy

7copy8copy

9copy

10copy

11copy12copy

13copy

NGLNG










84 Discussion 143



+ 6 CompositionNmax











84 Discussion









85 Conclusion




Bibliography 145



Bibliography











146 Bibliography













Chapter 9

Conclusion






147

148 Conclusion




Appendix A



A1 Introduction



149










Nominal conditions







NG

LC

SC

PC1 PC2

LNG


Ph

Ph

Ph

Pl

Pl

Pl

Pm

Pm

SWSW

SW

SW



































bull Pm in SC





Some remarks





Tem

pera

ture

[K]

0 50 100 150250

255

260

265

270

275

280

285

(a) NG1A


0 50 100 150215220225230235240245250255260

(b) NG1B


Position

Tem

pera

ture

[K]

0 50 100 150185

190

195

200

205

210

215

220

225

(c) NG2



120130140150160170180190200

(d) NG3









NG LNG


SWSW

SW

SW

TC

TCTCTC

PC

SHSHSH

SH













bull ToutNG1A

bull ToutNG1B

bull ToutNG2

bull Pm


A6 Conclusion


Bibliography 157



Bibliography






158 Bibliography

Appendix B







159


B1 Introduction









Eva

pora

tor13

Compressor13

Expansion Valve13

Con

dens

er13

Fan13Fan13


amb13T13

C13P13

CF13N13

C13N13E13P13

OD13

EF13N13

cabin13T13

e13Q13 13EF13W13 13

C13W13 13

CF13W13 13

Store13

Food13

s13c13Q13213 13

c13g13Q13213 13

food13T13





























ηis

hic =1minus fq






)





)






z )dt








15

2

25

3

35

456789101

15

2

25

PE (bar)

PC

(bar)

Wto

t (K

W)

A

B

Tcabin 1

Tcabin 2



0 6 12 18 240

5

10

15

20

25

30

35

Time [hr]

Qua

lity

loss

[]

Tfood

=2 C

Tfood

=Tsin2

Tfood

=Tsin1

Tfood

=1 C







J (B1)

where J =int t f


Wtot(t)

)dt (B2)












B4 Optimization

B41 Optimization













B4 Optimization 167





Cop =int year







75 80 85 90 95 100 105 110 11514

16

18

20

22

24

26

28


Dai

ly Q

ualit

y Lo

ss [

]

Case 1

Case 4

Case 3 Case 2



B5 Discussion



B6 Conclusion



Bibliography 169

NC

NCF

NEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF12 24 36 48 60 720

200400600800

100012001400

(b) Power

Time [h]

T[

C]

Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4





Bibliography


170 Bibliography

NC

NCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Inputs

Time [h]

W[W

] Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4








Bibliography 171

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]T

[C

]Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4







172 Bibliography

NCNCFNEF

Time [h]

N[r

ads-1

]

12 24 36 48 60 720

10

20

30

40

50

60

(a) Speed

Time [h]

W[W

]

Wtot

WC

WEFWCF

12 24 36 48 60 720200400600800

100012001400

(b) Power

Time [h]

T[

C] Tcon

Tamb

12 24 36 48 60 7210

20

30

40


Time [h]

T[

C]

Tevap

Tfood

Tcabin

12 24 36 48 60 72-8

-6

-4

-2

0

2

4



Contents

List of Figures

List of Tables

1 Introduction

11 Motivation

12 Thesis overview

13 Publication list


21 Introduction

211 Fundamentals


213 Condenser

214 Evaporator

215 Active charge






224 Cascaded cycles

23 Conclusion

Bibliography


31 Introduction






331 Optimal designs




342 Explanation


35 Discussion



36 Conclusion

Bibliography


41 Introduction


421 Linear analysis



431 Modelling



44 CO2 case study

441 Modelling



45 Discussion

451 Super-heating



46 Conclusion

Bibliography


51 Introduction








55 Discussion

551 Tmin-method



56 Conclusion

Bibliography


61 Introduction

611 Optimal design





64 Discussion

641 Compressor


643 Feed pressure


65 Conclusion

Bibliography


71 Introduction





73 Model







78 Discussion





Bibliography


81 Introduction




83 Case studies






84 Discussion




85 Conclusion

Bibliography

9 Conclusion


A1 Introduction









A6 Conclusion

Bibliography


B1 Introduction







B4 Optimization

B41 Optimization



B5 Discussion

B6 Conclusion

Bibliography

jørgen bauck jensen optimal operation of refrigera- tion

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