Design of a Cooling System for the Hybrid Engine Design of a Heat Exchanger for Cooling Bleed Air with Liqueﬁed Natural Gas

Home , Vapor quality

Design of a cooling system for the hybrid engine Design of a heat exchanger for cooling bleed air with liqueﬁed natural gas

K.G. Fohmann Technische Universiteit Delft

DESIGNOFACOOLINGSYSTEMFORTHE HYBRIDENGINE

DESIGNOFAHEATEXCHANGERFORCOOLINGBLEEDAIRWITH LIQUEFIED NATURAL GAS

K.G. Fohmann

in partial fulﬁllment of the requirements for the degree of

Master of Science in Aerospace Engineering

at the Delft University of Technology, to be defended publicly on Wednesday 12 August 2015 at 10:00am.

Student number: 4047680 Supervisor: Dr. A. G. Rao TU Delft Thesis committee: Prof. Dr. D.J.E.M. Roekaerts, TU Delft Dr. C.M. de Servi, TU Delft F.Yin, TU Delft

An electronic version of this thesis is available at http://repository.tudelft.nl/. Thesis registration number: 035#15#MT#FPP ii

DELFT UNIVERSITY OF TECHNOLOGY DEPARTMENT OF FLIGHT PERFORMANCEAND PROPULSION

The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineer- ing for acceptance a thesis entitled “Design of a cooling system for the hybrid engine - Design of a heat exchanger for cooling bleed air with liqueﬁed natural gas” by Karl Georg Fohmann in partial fulﬁllment of the requirements for the degree of Master of Science.

Dated: August 12, 2015

Chairman of thesis committee/Supervisor: Dr. A.G. Rao

Reader one Prof. Dr. D.J.E.M. Roekaerts

Reader two: Dr. C.M. de Servi

Reader two: F.Yin PREFACE

Soon in the course of my studies of Aerospace Engineering I found out, that my interest in this particular field was strongly focused on engines and their development. I was fascinated on how much energy could be extracted from a gaseous flow and also the simple principles on how that worked. Additional courses that I chose to build up my knowledge on gas turbines strengthened my wish to know more about these fascinating devices. Starting from this background, it was clear for me that this also would be the field in which I would like to graduate. The development of a heat exchanger for a gas turbine engine was a new area for me, since it is normally a mechanical engineering discipline. The combination with the requirements an aero gas turbine poses on the design gave rise to many interdisciplinary challenges that had to be faced during the process. The finished work can thus hopefully inspire colleagues to further dive into this interesting subject of future and more efficient engine designs and give them a starting point in their design ideas. I would like to take the opportunity to thank my supervisor Dr. A. G. Rao for his continuous support and for the advice he gave me throughout this work. Furthermore I would like to thank Dr. D. Dewanji for the support he gave me throughout the first half of my thesis, for which I am very thankful. A big help were also the advices I was given by Prof. Dr. D.J.E.M. Roekaerts for modeling the physical parameters. Also I would like to thank Dr. C. de Servi for his insights in heat transfer and Mrs. F.Yin for her support in the final steps of my thesis. Moreover I would like to thank TU Delft for giving me the possibility to perform this work. Finally I would very much like to thank my family, my father, mother, sister and uncle for their everlasting support of my studies in multiple fields. My thank also goes to my friends Hugo, Mirko, Daniel and Christo- pher for their company during my time in Delft or at home, for many nice hours, which formed an episode in my life which I will never forget. K.G. Fohmann Delft, July 28, 2015

iii

EXECUTIVE SUMMARY

Traveling has always been a major interest of humanity. It arose from the need to survive in a world that was continuously changing its environment. People had to travel to reach destinations that were more favorable for leading their lifes. Throughout the past many means of transportation have been developed to facilitate traveling and increase comfort. Nowadays the aircraft is the main mean of fast and economic transportation. Even though the motivation for traveling has changed throughout the centuries, traveling still enjoys a high degree of interest in society. To make this possible relentless effort is needed to make safe travel through the air possible and acquirable for every one. With the increasing awareness of the finite nature of fossil fuel reserves also the need to find alternative propellant becomes evident. One of the future challenges will therefore be to find alternative sources of fuel, that still can preserve our current status of comfort in traveling but also have a reduced impact on our environment. This master thesis aims at providing a contribution to lower the emissions made by future turbofan engines that are propelled with liquefied natural gas. This is done by designing a heat exchanger to exploit the deep temperatures of liquefied natural gas to lower the bleed air temperature required for cooling the high pressure turbine. To achieve a high thermal efficiency of the engine, a high turbine inlet temperature is favored. This temperature is limited by the melting temperature of the turbine blade material. To overcome this issue, cooling is applied. In current gas turbine engines, large amounts of bleed air are therefore used to cool the high pressure turbine. Due to the high pressures present in the high pressure turbine, the high pressure compressor offers the only source for cooling air. Since this bleed air can not be used for combustion, it does hardly contribute to the generation of thrust. It is therefore of interest to keep the requirements for cooling air as low as possible. By cooling down the bleed air from the compressor, the heat exchanger will reduce the mass flow required to cool down the high pressure turbine to the same temperature. This will increase the core mass flow and thus the efficiency of the overall engine. To be able to design a heat exchanger, fundamental knowledge in heat exchanger design, cryogenics, thermodynamics and gas turbine performance must be known. The heat exchanger will be designed for a hybrid engine developed in the ’Advanced Hybrid Engines for Air- craft Development’ program. Since only little data of this engine is known, most of the data will be taken from the General Electric GE-90-94B, which served as the hybrid engine’s basis. The heat exchanger will be designed by using semi-empirical models which will produce a preliminary design. The approach will focus on producing a thermal design of the exchanger. It turns out that the evaporation is a critical part of the design while the pressure drop for the configurations researched is of less importance. Based on the thermal calculations the heat exchanger is estimated to lower the specific fuel consumption of the hybrid engine by 0.68% at take-off and increases the thrust by 8.38%. Since the air mass flow remains the same, the specific thrust also will increase by 8.38%. The heat exchanger will be folded three times with a total length of 2.46m. It will have a diameter of 0.26m. Concluding it can be said that from this perspective applying the heat exchanger on gas turbines is beneficial to air transport.

CONTENTS

Preface iii Executive Summary v List of Figures xi List of Tables xiii Acronyms xv Latin symbols xvii Greek symbols xxiii 1 Introduction 1 1.1 Problem Statement...... 1 1.2 Purpose of Study...... 2 1.3 Consideration & Motivation...... 3 1.4 Objective & Goals...... 3 1.5 Contents...... 4 2 Literature Survey 5 2.1 Heat Exchanger...... 5 2.1.1 Fluid Analysis...... 5 2.1.2 Thermal Design Methods...... 9 2.1.3 Secondary effects...... 12 2.1.4 Type of heat exchangers...... 13 2.1.5 Mechanical Aspects...... 17 2.1.6 Single Phase Flow...... 18 2.1.7 Multiphase Flow...... 23 2.2 Turbine cooling...... 27 2.2.1 Convection Cooling...... 28 2.2.2 Jet impingement...... 30 2.2.3 Film Cooling...... 31 2.2.4 Blade material...... 33 2.2.5 Used Code...... 34 2.3 Engine performance...... 35 2.3.1 Calculations before engine intake...... 35 2.3.2 Calculations of combined airflow...... 36 2.3.3 Calculations of core gas flow...... 36 2.3.4 Choked Nozzle...... 38 2.3.5 Unchoked Nozzle...... 38 2.3.6 Calculations of bypass air flow...... 38 3 Module Implementation 41 3.1 Heat exchanger...... 43 3.1.1 General design considerations...... 44 3.1.2 Liquid phase flow...... 45 3.1.3 Two phase flow...... 48 3.1.4 Gas phase flow...... 48 3.1.5 Limitations...... 48 3.2 Turbine blade cooling module...... 51 3.3 Engine Performance...... 52

vii viii CONTENTS

4 Verification & Validation 55 4.1 Heat Exchanger...... 55 4.1.1 Verification...... 55 4.2 Engine Simulation...... 60 4.2.1 Verification...... 61 4.2.2 Validation...... 62 5 Results & Discussion 65 5.1 Inputs...... 65 5.1.1 Heat exchanger...... 65 5.1.2 Cooling...... 66 5.1.3 Engine Performance...... 66 5.2 Results from heat exchanger module...... 66 5.2.1 Heat transfer results...... 67 5.2.2 Phase change results...... 69 5.2.3 Pressure drop results...... 71 5.2.4 Flow parameter results...... 73 5.3 Results cooling module...... 74 5.4 Results engine performance module...... 75 5.5 Sensitivity analysis...... 77 6 Design Suggestion 79 7 Conclusions & Recommendations 81 7.1 Conclusions...... 81 7.2 Recommendations...... 82 Bibliography 85 A Two phase heat transfer correlations 89 A.1 Gungor-Winterton...... 89 A.2 Bertsch...... 90 A.3 Kandlikar...... 91 B Drawing of the heat exchanger 93 C Code 95 C.1 AccPressureloss...... 95 C.2 bend_pressuredrop...... 101 C.3 clearing...... 101 C.4 darcy_f...... 102 C.5 fin_temp_distr...... 102 C.6 flatratepressure...... 102 C.7 FricPressureloss...... 102 C.8 heatofvap...... 103 C.9 hex_run...... 104 C.10 imp_quan_pipe...... 107 C.11 imp_quan_shell...... 107 C.12 Layout_final...... 108 C.13 momentum_pressure_drop...... 109 C.14 Newpressure...... 110 C.15 Newpressure_air...... 110 C.16 overall_heatcoeff_sp...... 111 C.17 overall_heatcoeff_tp...... 112 C.18 PROPS_sp1...... 113 C.19 PROPS_sp2...... 113 C.20 Runfile...... 113 C.21 sat_temp...... 116 C.22 singlephase_heat_shell...... 116 C.23 singlephase_heat_tube...... 117 CONTENTS ix

C.24 singlephase_pressure_pipe...... 117 C.25 singlephase_pressure_pipe_lng...... 118 C.26 singlephase_pressure_shell...... 118 C.27 spphasetwo...... 119 C.28 totalPlot2...... 122 C.29 Pressureplot...... 124 C.30 tp_fric_frac...... 125 C.31 tp_heat...... 125 C.32 vapor_quality...... 127 C.33 vapor_quality_tool...... 128 C.34 void_fraction...... 128 C.35 wallT...... 128

LISTOF FIGURES

1.1 Conceptual layout of the AHEAD engine [1] ...... 2

2.1 Phase diagram (example of CO2), [2] ...... 7 2.2 a) Two phase flow regimes in a horizontal pipe and b)flow patterns during forced convection boiling in a pipe [3] ...... 8 2.3 Control volume for a heat exchanger [4] ...... 11 2.4 Effects to be considered for a given design effectiveness [5] ...... 12 2.5 a) Shell-and Tube exchanger with one shell and one tube pass b) exchanger featuring one shell pass and 2 tube passes [4] ...... 14 2.6 Gasketed plate and frame heat exchanger [4] ...... 15 2.7 Fin automotive condenser [4] ...... 17 2.8 a) individual finned tubes b) flat continuous fins on an array of tubes [4] ...... 17 2.9 The factor Kc and Ke as a function of σ [4] ...... 20 2.10 Rough and smooth friction factors for equation 2.50 [4] ...... 22 2.11 Kb∗ as a function of the bend, curvature and tube diameter [4] ...... 23 2.12 CRe as a function of Reynolds number [4] ...... 23 2.13 Cdev as a function of Length over diameter and Kb∗ [4]...... 24 2.14 Boiling phenomena and associated qualitative heat fluxes [6] ...... 24 2.15 Turbine inlet temperature increase over the decades [7] ...... 28 2.16 Cut through a convective cooling turbine blade [8] ...... 29 2.17 Cooling enhancement by a) ribs and b)pin fins [9] ...... 29 2.18 Film cooling - different heat zones ...... 32 2.19 Material development a) polycrystal b) directionally solidified c) single crystal [10] ...... 33 2.20 Starting layout of the Graphical User Interphase (GUI) of the code [11] ...... 34 2.21 Scheme of the hybrid engine [1] ...... 35

3.1 Solution procedure for the design problem ...... 42 3.2 Artiﬁcial shells (blue dotted lines) ...... 46 3.3 Calculation scheme of the heat exchanger module ...... 46 3.4 Liquid phase part of the heat exchanger module ...... 47 3.5 Two phase part of the heat exchanger module ...... 49 3.6 Gas phase part of the heat exchanger module ...... 50 3.7 Engine simulation layout ...... 52

4.1 Comparison between friction factor used by Moody and Mueller-Steinhagen and Heck . . . . . 57 4.2 Two-phase heat transfer coefﬁcients based on different correlations ...... 58 4.3 Results of the heat exchanger module ...... 59 4.4 Results by Mueller-Steinhagen and Heck method ...... 59 4.5 Void fraction over vapor quality ...... 60

5.1 Cross sectional view of the heat exchanger ...... 67 5.2 Temperature distribution and heat transfer along the heat exchanger ...... 68 5.3 Heat transfer coefficients along the heat exchanger ...... 69 5.4 Heat transfer of the fins ...... 70 5.5 Vapor quality and void fraction along the exchanger ...... 70 5.6 Heat flux and superheat temperature along the two phase flow region ...... 71 5.7 Pressure drop within the two phase Liquefied Natural Gas (LNG) flow ...... 72 5.8 Pressure drop along the heat exchanger ...... 73

xi xii LISTOF FIGURES

5.9 Distribution of Reynolds number a), Prandtl number b), ﬂow speed c), and density d) along the heat exchanger ...... 74 5.10 Change in thrust with bleed air ...... 76 5.11 Change in Speciﬁc Fuel Consumption (SFC) with bleed air ...... 76 5.12 Cross sectional operational envelope ...... 77 5.13 Design envelope and effectivenesses ...... 78

6.1 Design suggestion for the heat exchanger ...... 79

B.1 Test ...... 94 LISTOF TABLES

2.1 Mole percentage of LNG constituents from different plants [12],[13] ...... 6 2.2 LNG composition, averaged [12] [13] ...... 6 2.3 Air composition [14] ...... 6 2.4 Effectiveness and Number of Transfer Units (NTU) for different applications [4] ...... 13 2.5 Validity range for wall temperature calculation ...... 25 2.6 Coefﬁcients for Nusselt number determination from Florschuetz(1981) [15] ...... 30

3.1 Fluid parameters for the design ...... 43 3.2 Leading edge cooling parameters ...... 51 3.3 Centerline cooling parameters ...... 51 3.4 Trailing edge cooling parameters ...... 51 3.5 Engine operating conditions at design point GE-90-94 [16] ...... 52 3.6 Engine operating conditions at off design point GE-90-94B ...... 53 3.7 Engine operating conditions at static sea level GE-90-94B ...... 53

4.1 Input parameters overall heat transfer coefficient verification, example 23-1, [17] ...... 56 4.2 Result comparison overall heat transfer coefficient verification, [18] ...... 56 4.3 Inputs for evaluating the different heat transfer correlations ...... 57 4.4 Input parameters frictional pressure drop verification, [19] ...... 59 4.5 Design condition engine verification GE-90-94AB, [18] ...... 61 4.6 Design condition engine verification GE90, [16] ...... 61 4.7 Off-design condition engine verification GE90, [16] ...... 62

5.1 Fluid input requirements ...... 65 5.2 Fluid input requirements ...... 66 5.3 Turbine parameters ...... 66 5.4 Heat exchanger geometrical speciﬁcations ...... 67 5.5 Cooling module results - mass ﬂows ...... 75 5.7 SFC comparison of engine with and without heat exchanger ...... 75 5.6 Thrust comparison of engine with and without heat exchanger ...... 75

A.1 Coefﬁcients for the Kandlikar relation ...... 91 A.2 Fluid dependent parameter for the Kandlikar relation ...... 91

xiii

ACRONYMS

AHEAD Advanced Hybrid Engines for Aircraft Develop- ment

CFD Computational Fluid Dynamics CHF Critical Heat Flux

DNB Departure from Nucleate Boiling

EOS Equation of State

FOD Foreign Object Damage

GERG Groupe Européen de Recherches Gazières GSP Gas Turbine Simulation Program GUI Graphical User Interphase

HEx Heat Exchanger HPC High Pressure Compressor HPT High Pressure Turbine

I/O Input/Output ISO International Organization for Standardization

LH2 Liquid Hydrogen LHV Lower Heating Value LMTD Logarithmic Mean Temperature Difference LNG Liqueﬁed Natural Gas LPC Low Pressure Compressor

MTD Mean Temperature Difference

NGV Nozzle Guide Vane NIST National Institute for Standardization and Tech- nology NTU Number of Transfer Units

ODS Oxide Dispersion Strengthened

PFHE Plate-ﬁn Heat Exchanger PMBOK Project Management Body of Knowledge

SFC Speciﬁc Fuel Consumption

TBC Thermal Barrier Coating TIT Turbine Inlet Temperature TSFC Thrust Speciﬁc Fuel Consumption

LATINSYMBOLS

2 Ak Cross-sectional area for conduction [m ] A Cross-sectional area [m2] A Heat transfer surface area [m2] A Parameter A for jet impingement heat transfer [ ] − a Parameter a Bertsch [ ] − 2 Ach,in Channel inlet area for jet impingement [m ] 2 ACR Critical area [m ] 2 Ain Nozzle inlet area for jet impingement [m ] 2 Anoz Nozzle area [m ] A Pool boiling parameter Liu Winterton [ ] p − B Parameter B for jet impingement heat transfer [ ] − b Parameter b Bertsch [ ] − Bo Boiling number [ ] − C Intermediate constant Liu Winterton [ ] − C Boundary condition constant for fin calculation [ ] − C Parameter C for jet impingement heat transfer [ ] − C ∗ Heat capacity rate ratio [ ] − C1 Constant Kandlikar [ ] − C2 Constant Kandlikar [ ] − C3 Constant Kandlikar [ ] − C4 Constant Kandlikar [ ] − C5 Constant Kandlikar [ ] − Cc Heat capacity rate of cold fluid [W /K ] Cdev Correction factor for flow development after a bend [ ] − Ch Heat capacity rate of hot fluid [W /K ] Cmax Maximum of Cc or Ch [W /K ] Cmin Minimum of Cc or Ch [W /K ] Co Confinement number Bertsch [ ] − cp Specific heat at constant pressure [J/kg/K ] cp,air Specific heat at constant pressure, Air [J/kg/K ] cp,g as Specific heat at constant pressure, Gas [J/kg/K ] cp,g asmi x1 Specific heat at constant pressure, Gas-mix1 [J/kg/K ] cp,g asmi x2 Specific heat at constant pressure, Gas-mix2 [J/kg/K ] cp,min Minimum cp of the two fluids [J/kg/K ] CRe 0 Correction factor for Reynolds number evaluated at r /d 1 [ ] c i = − C Correction factor for Reynolds number [ ] Re − C Correction factor for surface roughness [ ] r ough − cv Specific heat at constant volume [J/kg/K ] D Jet hole diameter for jet impingement pressure drop[m] d Diameter [m] d Jet hole diameter for jet impingement heat transfer [m] Dh Hydraulic diameter [m]

xvii xviii LISTOF TABLES

³ ´ dp Frictional pressure drop parameter, liquid [Pa] dL f ,l ³ ´ dp Frictional pressure drop parameter, two phase dL f ,tp [Pa] ³ ´ dp Frictional pressure drop parameter, vapor [Pa] dL f ,v dx Elemental discretization step [m] g Gravitational acceleration [m/s2] E Enhancement factor Gungor Winterton [ ] − E2 Enhancement factor addition Gungor Winterton [ ] − Fr Froude number [ ] − F Enhancement factor Bertsch [ ] − F Log mean temperature difference correction factor [ ] − F Enhancement factor Liu Winterton [ ] − F Thrust [kN] f Darcy friction factor [ ] D − f Fanning friction factor [ ] F − F Fluid dependent parameter Kandlikar [ ] f l − f Rough Fanning friction factor [ ] F,r ough − f Smooth Fanning friction factor [ ] F,smooth − Frlo Froude number with all mass flow treated as liquid [ ] − Ftot Total thrust [kN] G Fluid mass velocity based on minimum free area [kg/(m2s)] G Mass flux [kg/(m2s)] Gc Cross flow velocity [m/s] gc Proportionality constant in Newton’s 2nd law of motion [ ] − G j Jet velocity [m/s] h Heat transfer coefficient [kW /m2/K ] hconv,l Liquid forced convection heat transfer coefficient [W /m2.K ] hconv,tp Two phase convective heat transfer coefficient [W /m2.K ] hconv,v Vapor forced convection heat transfer coefficient [W /m2.K ] 2 hFB Flow boiling heat transfer coefficient [W /m .K ] 2 h f in Fin heat transfer coefficient [W /(m K )] 2 hi Inner heat transfer coefficient [W /(m K )] hL Heat transfer coefficient with entire mass flow treated as liquid [W /m2.K ] 2 hl Liquid heat transfer coefficient [W /m .K ] hnb Nucleate boiling heat transfer coefficient [W /m2.K ] 2 ho Outer heat transfer coefficient [W /(m K )] 2 hpool Pool boiling heat transfer coefficient [W /m .K ] 2 hsp Single phase heat transfer coefficient [W /m .K ] 2 htp Two phase heat transfer coefficient [W /m .K ] i Flow arrangement specifier [ ] − K Pressure drop coefficient due to curvature [ ] b − K ∗ Pressure drop coefficient due to curvature for Re b = 106 [ ] − K Total pressure drop bend coefficient [ ] b,t − LISTOF TABLES xix

Kc Contraction loss coefficient at entrance of heat exchanger [ ] − Ke Contraction loss coefficient at exit of heat exchanger [ ] − K Pressure drop coefficient due to friction [ ] f − k Thermal conductivity fluid [W /m K ] f · k Thermal conductivity fin material [W /m K ] f in · k Thermal conductivity [W /m K ] t · k Thermal conductivity liquid phase[W /m K ] t,l · L Total length [m] l Fin length [m] LCV Lower calorific value [J/kg] LHV Lower Heating Value [J/kg] M Mach number [ ] − M Blowing ratio [ ] − m˙ Mass flow rate [kg/s] m Fin parameter [1/m] m Reynolds number exponent for jet impingement heat transfer [ ] − m˙ 2 Mass flow rate, station 2 [kg/s] m˙ 21 Mass flow rate, station 21 [kg/s] m˙ 25 Mass flow rate, station 25 [kg/s] m˙ 3 Mass flow rate, station 3 [kg/s] m˙ 31 Mass flow rate, station 31 [kg/s] m˙ 31.5 Mass flow rate, station 31.5 [kg/s] m˙ 32 Mass flow rate, station 32 [kg/s] M Mass [kg] m˙ air Mass flow rate air [kg/s] m˙ bleed Bleed mass flow rate[kg/s] m˙ cool,air Cooling mass flow rate [kg/s] m˙ cool,NGV NGV coolant mass flow rate [kg/s] m˙ cool,Tur b Turbine coolant mass flow rate [kg/s] m˙ cor r Corrected mass flow rate [kg/s] m˙ f Mass flow rate fuel [kg/s] m˙ i Internal mass flow rate [kg/s] m˙ LNG Mass flow rate LNG [kg/s] M Molecular weight [g/mol] m˙ tot Total mass flow rate [kg/s] n Exponent in cross flow function for jet impingement heat transfer [ ] − No Number of fins per tube [ ] f ins − Nu Nusselt number [ ] − nx Parameter nx for jet impingement heat transfer [ ] − ny Parameter ny for jet impingement heat transfer [ ] − n Parameter n for jet impingement heat transfer [ ] z z − P Temperature effectiveness [ ] − P Wetted perimeter [m] P Pressure [Pa] pamb Ambient pressure [Pa] pcool Coolant pressure [Pa] pCR Critical pressure [Pa] Pr Prandtl number [ ] − p Reduced pressure [ ] r − PR Fan pressure ratio [ ] f an − xx LISTOF TABLES

Pr Prandtl number liquid phase [ ] l − pt,2 Total pressure, station 2 [Pa] pt,21 Total pressure, station 21 [Pa] pt,31 Total pressure, station 31 [Pa] pt,31.5 Total pressure, station 31.5 [Pa] pt,32 Total pressure, station 32 [Pa] pt,5 Total pressure, station 5 [Pa] q Heat transfer rate [W ] q0 Conductive heat transfer rate at fin base [W ] q Heat flux [W /m2] qconv Convective heat transfer rate [W ] qL Heat flux with all mass flow treated as liquid [W /m2] qmax Maximum possible transfer rate [W ] R Heat capacity rate ratio [ ] − R Universal gas constant [J/kg K ] · r Radius [m] Re Reynolds number [ ] − Re Jet Reynolds number [ ] j − Re Reynolds number liquid phase [ ] l − Re Reynolds number vapor phase [ ] v − rh Hydraulic radius [m] ri Inner radius [m] ro Outer radius [m] R Surface roughness parameter Bertsch [ ] P − S Suppression factor Liu Winterton [ ] − S Suppression factor Gungor Winterton [ ] − S Suppression factor addition Gungor Winterton [ ] 2 − S Suppression factor Bertsch [ ] 2 − T Temperature [K ] t Sheet thickness [m] T Temperature of fin base [K ] T1 Temperature of fluid 1 [K ] T2 Temperature of fluid 2 [K ] TAir Temperature of air [K ] Tamb Ambient temperature [K ] Tc,i Cold stream inlet temperature [K ] Tc,o Cold stream outlet temperature [K ] Tcool Coolant temperature [K ] TCR Critical temperature [K ] T f ilm Cooling film temperature [K ] T f ree Free stream temperature [K ] Th,i Hot stream inlet temperature [K ] Th,o Hot stream outlet temperature [K ] TLNG Temperature of LNG [K ] Ts Static temperature [K ] Ts Saturation temperature [K ] Tt,2 Total temperature, station 2 [K ] Tt,21 Total temperature, station 21 [K ] Tt,25 Total temperature, station 25 [K ] Tt,3 Total temperature, station 3 [K ] Tt,31 Total temperature, station 31 [K ] Tt,31.5 Total temperature, station 31.5 [K ] Tt,32 Total temperature, station 32 [K ] Tt,5 Total temperature, station 5 [K ] Tt,7 Total temperature, station 7 [K ] LISTOF TABLES xxi

Tw Wall temperature [K ] U Overall heat transfer coefficient [W /(m2K )] u Flow velocity [m/s] um Mean flow velocity [m/s] V8 Velocity, station 8 [m/s] V Velocity [m/s] V Free stream velocity [m/s] ∞ Vc Coolant flow velocity [m/s] Vch,out Channel outlet flow velocity [m/s] 3 v f Specific volume of saturated fluid [m /kg] 3 vg Specific volume of gas [m /kg] Vi,n Nozzle internal velocity [m/s] Vj Jet velocity [m/s] Wbooster Work done by the booster [W ] Wf an Work done by the fan [W ] Whpc Work done by the HPC [W ] x Position/Length [m] x˙ Flow quality [ ] − x Thermodynamic equilibrium quality [ ] e − xe,in Thermodynamic equilibrium quality, sectional inlet [ ] − xe,out Thermodynamic equilibrium quality, sectional outlet [ ] − xn Streamwise jet hole spacing for jet impingement heat transfer [ ] − X Martinelli parameter [ ] tt − yn Spanwise jet hole spacing for jet impingement heat transfer [ ] − z Channel height for jet impingement heat transfer [m]

GREEKSYMBOLS

α Void fraction [ ] − α Void fraction at section inlet [ ] in − α Void fraction at section outlet[ ] out − δ Pressure ratio [ ] − ∆H Latent heat of vaporization [k J/kg] ∆p Pressure drop [Pa] ∆p A Pressure drop due to change in cross-sectional area [Pa] ∆pa Pressure drop due to acceleration [Pa] ∆pbend Bend pressure drop [Pa] ∆pch Crossflow pressure drop jet impingement [Pa] ∆pch,out Pressure drop due to channel outlet, jet impingement [Pa] ∆pcore Core pressure drop [Pa] ∆pe Exit pressure drop [Pa] ∆p f Frictonal pressure drop [Pa] ∆p f ,tp Frictional pressure drop in two phase flow per section [Pa] ∆pi Inlet pressure drop [Pa] ∆pmom Pressure drop due to change in momentum [Pa] ∆pn Pressure drop components jet impingement [Pa] ∆pn,f r Pressure drop due to nozzle friction, jet impingement [Pa] ∆pn,in Pressure drop due to nozzle inlet, jet impingement [Pa] ∆pn,out Pressure drop due to nozzle outlet, jet impingement [Pa] ∆ptot Total pressure drop [Pa] ∆ptot,tp Total two phase pressure drop [Pa] ∆T1 Temperature change across one section of fluid 1 [K ] ∆T2 Temperature change across one section of fluid 2 [K ] ∆TI Temperature difference between hot inlet and cold outlet [K ] ∆TII Temperature difference between hot outlet and cold inlet [K ] ∆Tlm Logarithmic Mean Temperature difference [K ] ∆Tm Temperature difference [K ]

∆Tmax Inlet temperature difference [K ] ∆Tsu Superheat temperature [K ] ∆x Elemental discretization step [m] ² Critical pressure ratio [ ] CR − ² Heat exchanger effectiveness [ ] − η Effectiveness [ ] − η Efficiency combustion chamber[ ] cc − η Isentropic efficiency [ ] i so − η Mechanical efficiency[ ] mech − η Nozzle efficiency[ ] i s,noz −

xxiii xxiv LISTOF TABLES

κ Specific heat ratio c /c [ ] p v − κ Specific heat ratio of air c /c [ ] air p v − κ Specific heat ratio of flue gas c /c [ ] g as p v − λ Latent heat of vaporization [J/kg] µ Liquid dynamic viscosity [Pa s] l · µ Vapor dynamic viscosity [Pa s] v · µ Dynamic viscosity [Pa s] · ν Specific volume [m3/kg] 3 νi Inlet specific volume [m /kg] 3 νm Mean specific volume [m /kg] 3 νo Outlet specific volume [m /kg] ψ Parameter Psi ∆T /(T T )[ ] m h,i − c,i − ρ Density [kg/m3] ρ Free stream density [kg/m3] ∞ 3 ρc Coolant density [kg/m ] 3 ρcr Critical density [kg/m ] 3 ρi Inlet density [kg/m ] 3 ρl Liquid sectional density [kg/m ] 3 ρm Mean sectional density [kg/m ] 3 ρo Outlet density [kg/m ] 3 ρv Vapor sectional density [kg/m ] σ Surface tension [N/m] σ Ratio of free flow area to frontal area (contraction ratio) [ ] − σ Stress [Pa] θ Measurement point value for wall temperature [ ] − θ Temperature ratio [ ] − θ Excess temperature [K ] θ0 Excess temperature for fin analysis [K ] ξn Nozzle pressure drop coefficient for jet impingement [ ] − ξn,f r Nozzle friction pressure drop coefficient for jet impingement [ ] − ξn,in Inlet pressure drop coefficient for jet impingement [ ] − ξn,out Exit pressure drop coefficient for jet impingement [ ] − ζ Friction factor, liquid [ ] l − ζ Friction factor, vapor [ ] v − 1

INTRODUCTION

Nowadays the topics of fuel saving and environmental friendliness are a dominant branch in research and academics. The finite nature of fossil fuels leads to the need to use the given resources efficiently and eventually find alternatives. The trend towards more and more fuel efficient power plants becomes apparent when regarding the cost share of fuel. Especially for aviation this is a very important part since the fuel costs are fixed costs and can count up to as much as a third of the total operation costs [20]. In the recent years the fuel costs have increased drastically. The share of fuel costs on total operating cost has even exceeded the share of labor costs of airlines. Therefore, it is of increased interest of airlines to operate efficient engines in order to be competitive. Often the choice of the engine is the only possibility to affect the fuel consumption of a given aircraft. For this reason, research in engine development is of utmost importance.

The general trend in the design of turbofan engines is the use of higher bypass ratios in order to reduce the relative share of the jet thrust. This increases the thermodynamic efﬁciency but leads to an increase in diameter of the engine fan. The theoretical size of the fan however is limited by operational and aerodynamic considerations. Therefore this approach will quickly face its limits. Further measures to reduce the fuel consumption of a turbofan incorporate the use of a gearbox for the fan in order to reduce fan tip speed. By doing so high blade tip velocities can be avoided and the accompanying aerodynamic losses can be lowered. Moreover the noise produced by the fan is reduced. This however increases complexity and maintenance efforts.

Engines in the far future will very likely operate on other fuels than the ones we use nowadays. For the bridg- ing time it is therefore of interest to have an engine being able to make use of fossil fuels as well as of future fuels. Two possibilities of such fuels are LNG or Liquid Hydrogen (LH2). These cryogenic fluids both show significant advantages in terms of CO2 emission and offer further benefits to be exploited. A potential benefit can be seen in the low temperature at which these fluids are stored. The central aspect of this thesis is therefore the preliminary thermal design of a heat exchanger making use of the low temperature of LNG. In modern turbofan engines the present temperatures already exceed the melting temperatures of metals by a far amount, for which reason turbines have to be cooled with bleed air from the compressor. Being the only possible solution to cool the turbine, bleed air is very expensive and therefore should be used economi- cally.

This thesis bases on the idea to cool down the bleed air coming from the compressor with the aid of the cryogenic fuel. Hereby the required amount of cooling air for the turbine is reduced. This increases the thermodynamic efﬁciency of the engine and reduces the amount of fuel to be burned. The heat exchange process will be performed by a heat exchanger.

1.1. PROBLEM STATEMENT

The more bleed air is extracted from the compressor, the more the thermal efﬁciency will decrease. This can be seen from the thermal efﬁciency relation given in equation 1.1. With more bleed air extracted the jet

1 2 1.I NTRODUCTION

Figure 1.1: Conceptual layout of the AHEAD engine [1] velocity will reduce. Thus the efﬁciency will decrease.

1 2 2 Σ 2 m˙ (Vj V ) ηther mal − ∞ (1.1) = m˙ f LHV

The efficiency decreases further due to the entropy generation that occurs when the cold bleed air stream mixes with the hot exhaust gases streaming in the turbine. Also the work needed to compress the air cannot be recovered by the turbine when it is used for cooling. This compressed air has little further beneficial effect on the performance output of the engine and thus has to be regarded as a loss. This is due to the irreversible effects accompanied by the expansion process of the bleed air when it is led through the cooling channels and finally back into the main gas stream. The more bleed air is needed, the higher the accompanying losses. This is a problem that has to be solved. Heat transfer plays a major role in this thesis. In equation 1.2 it can be seen how heat transfer works essentially. q h A ∆T (1.2) = · · The heat flux q is described by • The heat transfer coefficient h • The heat transfer surface area A

• The temperature difference ∆T This equation indicates the possibilities of changing the heat flux. The first one being the heat transfer coefficient. This parameter is a function of the flow parameters as well as the fluid parameters and the geometry of the fluid guiding channel. The heat transfer surface area A is to be seen from a macroscopic perspective. It is the overall area in the heat exchanger over which heat can be transferred from one fluid to the other. By altering the geometry this parameter can be influenced. Finally the temperature difference is the last measure to affect the heat flux. It however cannot be regarded as a design parameter since it is a requirement for the considered design. However by adjusting the material of the heat exchanger wall temperatures can be influenced.

1.2. PURPOSEOF STUDY

The developments done in this thesis are made for the Advanced Hybrid Engines for Aircraft Development (AHEAD) engine. A scheme of this engine can be seen in ﬁgure 1.1. This engine incorporates several new approaches to save fuel into a single engine. Among others also the alternative fuel question will be addressed. The engine will make use of conventional fuel as well as of a cryogenic fuel source like LNG or LH2. The purpose of this study aims to contribute a thermal design suggestion for a cryogenic heat exchanger in which heat is exchanged between the cryogenic fuel and bleed air. In this master thesis LNG will be used as the cryogenic ﬂuid. Moreover a gas turbine performance module will be developed to monitor the effect of the cooler bleed air on the engine performance. 1.3.C ONSIDERATION &MOTIVATION 3

In order to facilitate the process of designing the heat exchanger, a program will be created. This heat exchanger program will be kept simple to allow for quick adaptation in geometry or ﬂow arrangement. Since the output of this master thesis will be a preliminary design, the focus of the program will be set on ﬂexibility and speed. Thus, it will be possible to determine major effects of geometry changes on the outcome before a detailed and time consuming Computational Fluid Dynamics (CFD) simulation has to be run. Besides the fact that the primary goal is to give a thermal design suggestion for the heat exchanger the module can also be used easily for a redesign. To reach these goals correlations found in literature will be employed.

1.3. CONSIDERATION &MOTIVATION

Since the field of study is new there is a large uncertainty about the topic. In general the field of cryogenic heat exchangers at high pressures and temperatures is hardly researched. For aviation, this topic has not been researched yet or only locally. A hurdle can be seen in the phase change that will happen during the process. In this process LNG will evaporate and transform to its gaseous equivalent as which it then will be injected into the combustion chamber. This process involves a very high degree of nonlinearity and is hardly understood. Even though there are plenty of correlations to predict the phase change behavior, the majority of these models bases on empirical observations and is only valid in a very limited amount of fluids and operating conditions. Additionally the accuracy of those methods is generally rather poor. This roots in the turbulent nature of a phase changing process and the many parameters that influence the process. Therefore it is common to make assumptions in order to make the problem solvable. To be able to produce a design, fundamental research in the fields of cryogenics, heat exchangers, phase change behavior and gas turbine performance is necessary. This will help to develop a program with which the performance of a heat exchanger using LNG and air can be estimated. Such a program satisfies the need of an initial design tool for heat exchangers for evaporating applications ofLNG. In combination with a turbine blade cooling module as provided by [11] and a gas turbine performance module, it will be possible to assess the effect of such a heat exchanger on the overall gas turbine performance.

1.4. OBJECTIVE &GOALS

The objective of this master thesis is to provide a thermal design suggestion for a cryogenic heat exchanger for the AHEAD engine, which can later on easily be modiﬁed and adapted with the program. The design will be based on steady state calculations and valid for the full thrust take-off condition at sea level. The research question can be formulated as follows:

’Is it possible to assess the thermal performance of a cryogenic heat exchanger and its effect on the bleed air consumption of the cooling system of a gas turbine using semi empirical correlations?’

From this research question the following sub questions are derived: • What are the driving factors in a heat exchanger design? • What correlations are needed to model heat transfer and pressure drop in single and two phase ﬂow? • How will the temperature change of bleed air affect the amount of bleed air needed? • How much better will the engine perform based on thrust and SFC? From these questions sub goals can be derived: • Perform a literature survey to gain insight in heat exchanger design process • Retrieve parametric correlations from the body of knowledge to model the heat exchanger • Develop a model to assess the performance of the heat exchanger • Develop a model to assess the engine performance by using Gas Turbine Simulation Program (GSP) 4 1.I NTRODUCTION

• Determine the cooling need of the High Pressure Turbine (HPT) by making use of the code provided in [11] • Combine the tools to get an overall impression on the total performance and to be able to make a design suggestion

1.5. CONTENTS The chapters of this master thesis are as follows: Firstly in chapter 2 a discussion on the literature for this master thesis will be given. All the relevant fields will be addressed and explained in how far they are needed to fulfill the above stated goals. In chapter 3 the implementation of the methods discussed in 2 is given. Emphasis is laid on lining out the differences and the generic approach used. Next, chapter 4 will discuss the verification and validation processes to ensure validity of the results. In chapter 5 the results of the study will be shown and discussed in detail. After this, a design suggestion is made in chapter 6. Finally the master thesis will conclude with a conclusion and recommendations for future work in chapter 7. 2

LITERATURE SURVEY

This chapter will discuss the theory behind this work. It will discuss all the relevant physical phenomena needed for a preliminary thermal design. The level of detail will be kept to the required degree to model the heat exchanger for a preliminary design. This implies the use of semi empirical models and the required numerical implementation methods. Therefore this chapter puts emphasis on understanding the physics going on in the thermodynamic behavior of heat exchange and fluid wall interaction. This literature review will be sectioned into three modules. First of all the heat exchanger module will be covered. This module will be split up into different sections. Those will address topics such as an analysis of working fluids, single phase and two phase flow, pressure drop calculations, fins and thermal design methodologies. The second module to be covered will deal with the turbine blade cooling module that will be implemented based on the code of Tiemstra [11]. It will be split in a description of the physics that are modeled in the code as well as a discussion on the already existing code in general. The last section of this literature review will show the necessary calculations to determine thrust and SFC of a gas turbine engine at the design point. Based on this a discussion outlining the differences between design point calculations and off design point calculations will be given.

2.1. HEAT EXCHANGER

Heat exchangers exist in various forms and for many purposes. They are present in nearly every technical discipline and fulfill many different purposes. Also heat exchangers can be found in any environment possible. From heating systems in cold areas of the world where they provide warmth for people and animals to cli- mate controls in desert regions where heat exchangers cool down entire housing blocks. Heat exchangers are various in their design since they are mostly very purpose driven devices. This means, that heat exchangers often are custom designs, developed exactly to meet a specific purpose. This can reach from a house heating system over a car radiator to complex systems such as cooling systems for power plants and reactors. In all these applications heat exchangers work with fluids to be heated or cooled. For this reason a fluid analysis has to be performed before a suitable heat exchanger can be designed.

2.1.1. FLUID ANALYSIS In the given heat exchanger design a cryogenic fluid and air will exchange heat. For this special case LNG will be used since it will serve as fuel for the AHEAD engine. Another option would be LH2, which however will not be considered for this master thesis. This means that the fluids to be analyzed are both mixtures and not pure component fluids. Air consists mainly out of nitrogen and oxygen, while LNG consists mainly out of methane and ethane. Depending on where it is produced, the contents however may vary. This makes it harder to determine the exact behavior of LNG in terms of its thermodynamics. The major component of LNG is methane with around 95% of mole fraction [21]. Other sources state however, that among the different LNG plants, a partially big deviation in the content concentration exists [12], [13]. In table 2.1

5 6 2.L ITERATURE SURVEY

Table 2.1: Mole percentage of LNG constituents from different plants [12],[13]

Component Nigeria Arun Brunei Oman Atlantic Kenai Ras Laffan Das Islands Methane 87.9 88.48 89.40 90.00 95.00 99.80 90.28 84.50 Ethane 5.50 8.36 6.30 6.35 4.60 0.10 6.33 12.90 Propane 4.00 1.56 2.80 0.15 0.38 0.00 2.49 1.50 Butane 2.50 1.56 1.30 2.50 0 0 0.49 0.50 Nitrogen 0.10 0.04 0.20 1.00 0.03 0.10 0.41 0.60 Pentane 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00

the constituents from different LNG plants are given in mole percentage. Until now it has been of primary interest to model natural gas in its gaseous form. In the recent time with increase in demand of LNG, the interest in modeling LNG in its liquid phase and during its phase change has become bigger. This fosters research in this respective area. However until now (July 28, 2015) little information is known on the process of phase change of LNG. Phase change introduces a certain amount of complexity into the design. This results from the introduction of variables for properties which can be taken as constants for standard cases. Due to the cryogenic temperatures and the change in phase the transport and thermal properties of different substances cannot be regarded as constant. They are strongly dependent on the matter and its form. For gases e.g. a distinction has to be made between monoatomic and multi-atomic gases. This is because different mechanisms for storing energy within the molecule are used. For multi-atomic gases multiple energy storage principles like vibration, rotation and translation, are involved. Monoatomic gases on the other side only use translation to store energy. When and which energy mode is used for storing energy within the molecule is additionally dependent on temperature and pressure. Furthermore molecular forces and bonding forces can become important when determining the energy content of a molecule. Thus, very different behavior of different materials is given, which can have a substantial effect on cp and cv values as well as on thermal conductivity, viscosity and enthalpy. According to [3] it is most difficult to determine the actual state of energy mode in a liquid. When a liquid comes close to the freezing point, a lattice like structure can be found and energy storage occurs mostly in the form of vibration, whereas the amount of rotational energy is increased, when the substance is close the boiling point. When dealing with mixtures of different components it is even more difficult to predict the behavior of a substance. For this reason Equation of State (EOS) are invented to resemble the behavior of components. Those are usually of semi-empirical nature. To date however, the models to describe the physical properties are very complicated or lack of sufficient data is present in order to use them [22].

Still a problem is the behavior of LNG during the gasification process. Because of the difficulty of the involved processes, it is not possible to physically represent the phase change. Therefore one option can be seen in splitting LNG up into its constituents and calculate the phase change of each of them separately by making use of the percentage wise mass contribution to LNG. Increased interest is put on the specific heat of the mixture as well as thermal conductivity, density and enthalpy, since these quantities describe the thermodynamic behavior and also influence the heat transfer. An alternative way would be the modeling as a single mixture. In this option the influence of different constituents would describe the physical behavior of the mixture. This would facilitate most of the calculation processes, however the data used has to be very accurate. In tables 2.2 and 2.3, the average composition of LNG and air can be found respectively.

Table 2.2: LNG composition, averaged [12] [13] Table 2.3: Air composition [14]

Component LNG average, mole % Component Air average, mole % Methane 90.67 Nitrogen 78.08 Ethane 6.31 Oxygen 20.95 Propane 1.61 Argon 0.93 Butane 1.11 Carbon Dioxide 0.03 Pentane 0.00 Neon 0.00 Nitrogen 0.31 Methane 0.00 2.1.H EAT EXCHANGER 7

Figure 2.1: Phase diagram (example of CO2), [2]

For the calculations in the heat exchanger multiple properties of those ﬂuids are of importance. These are mainly the pressure, thermal abilities and phase change. The following properties have been identiﬁed to be of high importance for the design:

• Speciﬁc heat at constant pressure cp

• Thermal conductivity kt • Viscosity µ

• Latent heat of vaporization ∆H

• Density ρ

All these properties must be known in terms of pressure and temperature. Due to this requirement the Na- tional Institute for Standardization and Technology (NIST) database [23] was consulted to provide the necessary data. Since the concentration of the constituents of air as well as LNG are varying depending on location/source, the decision was made to limit the amount of constituents to make calculations possible. Es- pecially for LNG the reduction in components had to be done. The heat exchanger will operate at very high pressures and temperatures. Due to this some of the constituents of LNG were found in a state away from the vaporization boundary. If the pressure is higher than the critical pressure an increase in temperature will only lead to a supercritical state. A vaporization however cannot be performed. The only way how to avoid this is to reduce the pressure. This can be retraced regarding figure 2.1. It can be seen that for liquids in a region higher than the critical pressure no vaporization is possible unless the pressure is reduced below the critical pressure.To make calculations possible those substances were neglected. Fortunately the main components methane and ethane do not fall under this regime. A further problem arises during the phase change of LNG. The individual components will vaporize sooner or later during the evaporation, depending on their physical properties. This makes it difficult to assign the heat flow correctly to the individual fluid. This holds especially since during the phase change not all the volume is occupied by one phase but by two.

Additional concern arises from the pressure drop calculations. The methods presented in literature are usually valid only for pure components or mixtures in single phase state. For multi phase ﬂow with mixtures, data can be found for certain mixtures only, which are mostly refrigerants.

A very important aspect of the heat exchanger design will be the modeling of the phase change. The phase change will contribute significantly to the heat transfer rate of the heat exchanger. In this section only methods applicable to tube flow will be presented, since pool boiling is not suited for the application due to the large required boiling surface. Additionally, the mass flux is too high to allow for a static boiling process. Be- cause the mass flow rates of fuel vary with thrust setting, the LNG will start to boil at a different position and different speed in the heat exchanger. Since the requirement however was steady state condition, only the condition at take-off will be taken into account.

The flow phenomena occurring during a phase change cannot simply be assigned to either turbulent or to 8 2.L ITERATURE SURVEY laminar flow. During a phase change the fluid more or less quickly undergoes a sequence of different flow regimes. This is also very dependent on the guiding object through which the fluid is flowing. An example from [3] for different flow regimes in a horizontal pipe can be found in figure 2.2a).

Figure 2.2: a) Two phase ﬂow regimes in a horizontal pipe and b)ﬂow patterns during forced convection boiling in a pipe [3]

The ﬂow patterns may be described as follows:

1. Bubble Flow is occurring when the quality of the mixture is low and only few bubbles form

2. Plug ﬂow happens usually only in horizontal pipes. It occurs when the quality of the ﬂuid mixture is increased and the bubbles form together areas of vapor.

3. Stratified Flow is present when the flow velocities of both vapor and fluid are low and similar in mag- nitude. It can lead to thermal stress in the pipe due to unequal heat transfer rates.

4. Wavy Flow starts to occur when the ﬂow speed difference becomes bigger and shear forces gain in strength between the vapor and the ﬂuid.

5. Slug flow is a further stage in the process of increasing the speed of the vapor. In contrast to plug flow much more turbulence is involved in the flow behavior. Slug flow can be origin of vibrations.

6. Annular Flow occurs at even higher vapor flow velocities. The highly viscous fluid is being pressed on the outside of the wall and a ring-like pattern is formed. Droplets of liquid are found in the vapor stream. While in vertical pipes the annulus is evenly spread, horizontal pipes show more fluid mass at the bottom of the pipe due to the influence of gravity.

7. Mist flow is the counter principle of Bubble flow. The quality of the mixture is high and the liquid phase only exists in terms of droplets floating in the vapor flow.

These different ﬂow phenomena are very difﬁcult to describe physically. For this reason models have been created to resemble the physics of the phase change in an empirical manner. For this master thesis especially the heat transfer and pressure drop during the evaporation are relevant. These two quantities have to be found in order to provide a good heat exchanger design. In the following a discussion on how to calculate 2.1.H EAT EXCHANGER 9 pressure drop and heat transfer will be presented. As the basis for the analysis the works of Barron[3] and Smith[24] are taken.

2.1.2. THERMAL DESIGN METHODS

Heat exchangers have been already in use for a long time in the most various applications possible. Therefore to date there exist multiple design philosophies that are applied in engineering. Depending on the type of ﬂuids, the temperatures of the ﬂuids, the effectiveness required and the budget available various geometries exist. For this reason different design philosophies will be presented. They are called "Lumped parameter models" and share the same assumptions. According to Pacio et al. [5] these assumptions are the main reason why lumped parameter models cannot be taken as standalone solution for heat exchanger design. The main assumptions are listed in the following:

• Steady-state operating conditions

• No heat transfer with the surroundings

• Longitudinal heat conduction is negligible

• Constant overall heat transfer coefﬁcient

• Constant heat capacity

• Constant physical properties along the heat exchanger

Smith[24] identified two very basic approaches in designing a simple heat exchanger, namely rating and sizing. The same differentiation was obtained by Shah[4].While rating is concerned with finding the heat exchanger duty q when the heat exchanger geometry is given, sizing refers to finding a geometry for a given q. Both approaches cannot directly represent the design for this given case since in the beginning heat duty as well as geometry are unknown. Since however a classical design starts with the heat duty, the sizing process will be presented. This also goes along with the primary definition of "sizing" according to Shah[4]. To limit the discussion, the most frequently applied methods will be shown. Other, even older methods that exist but are not practically used anymore will therefore be omitted. The thermal design methods presented here are all based on a complete heat exchanger analysis. Therefore they can be used when no phase change is present. Furthermore they only give information about the thermal performance of a heat exchanger. Mechanical analysis or even life cycle estimations cannot be performed with this method. Since a further mechanical design is out of scope of this master thesis, a physical shape of the heat exchanger has to be qualitatively derived from the thermal results only. This mainly implies the translation from the heat transfer surface requirement to an actual shape and geometry.

THE MEAN TEMPERATURE METHOD

Smith[24] suggests two approaches to thermally design a unit heat exchanger. Those are the ’Logarithmic Mean Temperature Difference (LMTD)-NTU method’ and the ’²-NTU method’. To stay consistent in notation also with the other methods presented, the procedures, equations and figures used by Shah[4] will be taken. The first task according to Smith is to determine the mean temperature difference for heat transfer between the two fluids, as in equation 2.1. q UA∆T (2.1) = m LMTD stands for the Logarithmic Mean Temperature Difference and is defined by relation 2.2.

(∆TI ∆TII ) LMTD ∆Tlm − (2.2) = = ln(∆TI /∆TII )

In the case of a non parallel flow heat exchanger ∆TI is defined as depicted in relation 2.3 and ∆TII as shown in equation 2.4. ∆T T T (2.3) I = h,i − c,o ∆T T T (2.4) II = h,o − c,i 10 2.L ITERATURE SURVEY with the subscript h=hot, c=cold, i=inlet and o=outlet. For the arrangement of a counterflow or parallel flow heat exchanger as well as C ∗=0 exchangers, the log mean temperature difference is the same as the effective temperature difference T m, ∆T ∆T (2.5) m = lm For all other arrangements, a correction factor F is being introduced. It is defined in equation 2.6.

∆Tm q F (2.6) = ∆Tlm = UA∆Tlm The factor F is dependent on the temperature effectiveness P, the heat capacity rate ratio R and the flow arrangement. The factor of F should not be seen as a direct performance criterion but the degree to which the performance of the heat exchanger is lower than a counterflow heat exchanger. A factorF close to 1 gives therefore a heat exchanger performance close to the one of a counter flow type. As an additional example the F factor for a parallel flow type heat exchanger is given in equation 2.7. For evaporating fluids or condensing vapors in which the temperature is constant, ∆T ∆T ∆ and F =1. m = lm = Tmax (R 1)ln[(1 RP)/(1 P)] F + − − (2.7) = (R 1)ln[1 P(1 R)] − − + The LMTD method is used mainly for design of Shell-and-Tube heat exchangers. The NTU is defined as

UA NTU (2.8) = mc˙ p,min for fluids with constant heat transfer coefficient U. The additional parameters represent the transfer surface area A, the mass flow m˙ and cp as the constant specific heat. It has to be mentioned that at cryogenic temperatures the heat transfer coefficient can change significantly. Therefore it has to be taken as a variable for the required heat exchanger design. For sizing methods as discussed above, in the LMTD-NTU method, the energy balances are used directly to determine the unknown temperatures. The parameters of the overall thermal conductance U A can then be used for direct sizing. ·

THE ²-NTUMETHOD Shah complements the LMTD-NTU method by adding the ²-NTU method, the P-NTU method, the Mean Temperature Difference (MTD) method as well as the ψ-P and the P P method [4]. To stay compact only 1 − 2 the ²-NTU method will be presented here. The ²-NTU method gives the heat transfer rate in terms of a heat exchanger effectiveness ² and is deﬁned as:

q ² C (T T ) ² C ∆ (2.9) = · min h,i − c,i = · min Tmax

The effectiveness parameter is dependent on the NTU, the heat capacity rate ratio C ∗ as well as on the flow arrangement. It is a dimensionless parameter that can be defined as depicted in equation 2.10. It can be seen that the theoretical maximum value of ² can never exceed 1. The value of qmax is obtained from a counterflow heat exchanger featuring infinite heat transfer surface area while making use of the same mass flow rates, temperatures and pressures as the actual heat exchanger. It is however assumed for this "perfect" exchanger, that no heat leakage or longitudinal wall heat conduction is present. The effectiveness should be clearly separated from the term of efficiency, which represents either an energy conversion or a comparison of a system to its ideal performance [4].

q UA Cmin ² f ( , , flowarrangement) (2.10) = qmax = Cmin Cmax where Cmin refers to the heat capacity rate of the ﬂuid with the lower heat capacity rate and Cmax to the ﬂuid with the higher heat capacity rate. Together they form the heat capacity rate ratio as depicted in relation 2.11. Cmin (mc˙ p )min C ∗ (2.11) = Cmax = (mc˙ p )max

It should be aimed for a value of C ∗=1, in order to obtain a balanced design. It can be seen, that the heat capacity rate ratio is directly dependent to the mass flow rates of the two streams. Therefore it is a design 2.1.H EAT EXCHANGER 11 parameter that can be chosen by the design engineer to adjust the system. However it has to be mentioned that in case of an evaporating pure fluid or condensing pure fluid the ideal temperature change along the side of the evaporating/condensing fluid is zero. Since not always the mass flow rate can be adjusted, changing the heat transfer surface area offers a further method of adjusting the fluids to each other. In heat exchangers that use boiling or condensation additional effects come into play which can lead to a design which would not be optimal with respect to the heat capacitances. The discussion based on what the effectiveness for the heat exchanger will be leads to a geometry dependent answer. For the ²-NTU method tables are available that show the most common relations between ² and NTU. For a counterflow heat exchanger the effectiveness can be determined from formula 2.12, while the one for parallel flow can be retrieved from relation 2.13.

1 exp[ NTU(1 C ∗)] ² − − − (2.12) = 1 C exp[ NTU(1 C )] − ∗ − − ∗

1 exp[ NTU(1 C ∗)] ² − − + (2.13) = 1 C + ∗ In general the effectiveness can be displayed for any conﬁguration according to relations 2.14 and 2.15

Ch(Th,i Th,o) ² − (2.14) = C (T T ) min h,i − c,i

Cc (Tc,o Tc,i ) ² − (2.15) = C (T T ) min h,i − c,i With these main equations known and the help of some intermediate deﬁnitions, it is possible to determine thermodynamic data like heat transfer rate, in- and outlet temperatures as well as wall temperatures of a standard heat exchanger.

MODELING ACCORDING TO THE FIRST LAW OF THERMODYNAMICS

In contrast to the previous methods stands a direct modeling after the ﬁrst law of thermodynamics. This method, also known as heat balance method, has the advantage that it can be discretized easily. Furthermore it is a method that relies more on physical effects than on empirical knowledge. In contrast to the methods mentioned above, the energy balance method makes use of a control volume within the heat exchanger along which the physical phenomena are determined. An example of this can be found in ﬁgure 2.3. For a

Figure 2.3: Control volume for a heat exchanger [4] pure counterflow heat exchanger, the energy balance across one section can be found in relation 2.16 for the first fluid and in formula 2.17 for the second fluid. In this case i indicates the flow arrangement (parallel or contraflow, indicated by -1 or +1). µ ¶ dT1 i(mc˙ p )1T1 i(mc˙ p )1 T1 dx U(T1 T2)d A 0 (2.16) − + dx − − =

µ ¶ dT2 i(mc˙ p )2T2 i(mc˙ p )2 T2 dx U(T1 T2)d A 0 (2.17) − + dx + − = 12 2.L ITERATURE SURVEY

Derived from these two expressions, the change in temperature can be calculated that is obtained over the control volume section. In equation 2.18 and 2.19 the respective differential terms are given.

dT1 UA 1 (T2 T1) (2.18) dx = (mc˙ p )1 L −

dT2 UA 1 (T1 T2) (2.19) dx = i(mc˙ p )2 L −

With these equations the change in temperature can be determined over one control volume section. When this analysis is repeated for the entire length of the heat exchanger, the ﬁnal output temperature can be obtained. Since this method is very repetitive, it is used most efﬁciently when applied in a numerical scheme on a computer.

2.1.3. SECONDARY EFFECTS

In his article on thermal hydraulic models, Pacio [5] discusses further important phenomena that are usually neglected in heat exchanger design. These phenomena can be important to consider when it comes to cryogenic heat exchanger design. The most dominant simplifications made are ignoring the changes in fluid properties, heat leakage, flow distortion and thermal longitudinal heat conduction within the wall. The article also gives a quantitative estimation on the expected error that is produced introducing these assumptions. This is shown in a chart which can be found in figure 2.4.

Figure 2.4: Effects to be considered for a given design effectiveness [5]

The change in fluid properties was already treated by Barron[3]. This problem will be addressed by making use of thermodynamic data obtained from NIST. Flow distortion arises due to imperfections in the imple- mentations of the design in reality and due to geometric reasons. It represents a part of the flow that does not follow the implied route and therefore deteriorates the heat exchanger performance. Longitudinal heat conduction occurs in the walls of the tubes of the heat exchanger. Heat travels by conduction through the heat exchanger wall and therefore influences the heat release and absorption at another location. Ultimately heat leakage represents the heat exchange to the surrounding of the heat exchanger. This has to be counted as a loss. These losses can significantly impact the heat exchanger effectiveness obtained by a design. To get an impression on the effectivenesses of heat exchangers the value of NTU can be an indication. In this sense it can be seen as a direct measure of effectiveness, assuming the fluids and mass flow rates stay constant. Shah[4] gives some approximate values for the η-NTU relations for different applications. Those can be found in table 2.4. 2.1.H EAT EXCHANGER 13

Table 2.4: Effectiveness and NTU for different applications [4]

Application NTU ² Car radiator 0,5 40 % Steam plant condenser 1 63 % Regenerator for industrial gas turbine engine 10 90 % Regenerator for stirling engine 50 98 % Regenerator for LNG plant 100 99 %

From a qualitative comparison in size of these heat exchangers, it becomes clear that the heat exchanger for the AHEAD engine will not offer an effectiveness above 80%, also due to a mass and volume restriction for this exchanger.

Due to the purpose of the system which is to cool the compressor air the problem of heat leakage will not be addressed. This was decided, as hardly any isolation will be used anyway and heat loss in the bleed air to the surrounding air is highly appreciated. Thus not taking into account heat leakage will not harm the systems performance. Furthermore the estimation in ﬁgure 2.4 clearly shows, that the topic is relevant for heat exchanger effectivenesses from 95 % and above on. Hence this effect will not be taken into account.

The same argument is used when it comes to the longitudinal heat conduction in a wall. In a heat exchanger there is usually a temperature gradient present over the entire operating length. Due to this, also a temperature gradient will be present in the walls of the heat exchanger. Additionally conduction within the wall can influence the temperature of the fluids in a negative way. Logically the influence of this effect decreases with increased mass flow rates and smaller overall surfaces. Having a thin heat transfer surface furthermore reduces this effect. Mechanical limitations dictate however that in reality there will always be a certain portion of longitudinal heat conduction present in a heat exchanger.

2.1.4. TYPE OF HEAT EXCHANGERS The ways in which heat exchangers are designed are manifold. The designs vary in their geometry and material according to the requirements for temperature, the pressure, the operating ﬂuids, the required effectiveness, weight, volume or corrosion resistance. This section will introduce the main geometric concepts available and outlines their strengths and weaknesses in order to be able to choose a type of heat exchanger for the design. The different types of heat exchangers will be presented according to their geometrical and constructional differences. The following main heat exchangers are commonly named [25], [4]:

1. Tubular

2. Plate-type

3. Extended surface

4. Regenerators

The list does not claim to be complete. Since the variety of exchangers is so large only the most common ones were selected. Furthermore regenerators are not considered due to their apparent disadvantage in their discrete working behavior.

TUBULAR HEAT EXCHANGERS

Tubular heat exchangers are the most common heat exchangers available. They are usually designed as prime surface heat exchangers, having no heat transfer surfaces available, that do not transport any mass like e.g. ﬁns. The properties of tubular heat exchangers are generally as follows:

• High operating pressures

• High pressure difference between ﬂuids

• High operating temperature difference 14 2.L ITERATURE SURVEY

• Good fouling resistance • High heat exchange surface area translates usually into high volume As a sub category shell-and-tube, double-pipe and spiral-tube exchangers can be separated from each other [4]. Shell-and-tube heat exchangers are very common in industry. Their versatile and adjustable shape makes them a good choice for a variety of applications. An example of a shell-and-tube exchanger is depicted in figure 2.5. As the name indicates, this exchanger consists out of a shell, in which a bundle of tubes is mounted. One of the operating fluids flows through the tubes, while the second fluid flows in the compartment between the shell and tubes. To adjust the flow over the tubes, baffles, sheets or fins can be used. The ends of the typical shell-and-tube heat exchanger is usually covered with a front and a rear head, containing the flanges for mounting the heat exchanger to outside tubing. In many heat exchangers it is relatively easy to dismount the tube bundle from the shell for cleaning purposes.

Figure 2.5: a) Shell-and Tube exchanger with one shell and one tube pass b) exchanger featuring one shell pass and 2 tube passes [4]

The double-pipe heat exchanger is a pipe in pipe design. While one fluid flows between the two pipes, the other fluid flows in the inner pipe. Often those designs are combined with a U-shape in order to save space. Mostly double-pipe arrangements are used only for small-capacity applications in the range of up to 50m2 of heat transfer area. The difference to the shell-and-tube exchanger can be found mainly in the flow arrangement. In double pipe exchangers a counterflow or parallelflow arrangement can be found, while in shell-and-tube exchangers cross flow arrangements are most common. Ultimately the spiral tube design consists out of one or more layers of helically wound spiral coils in a shell. In these units a considerable surface area can be achieved and the heat transfer rates can be higher than for those of a double-pipe exchanger. However it has to be taken into account that the pressure drop in those exchangers can be substantial and that it is very hard to clean them.

PLATE-TYPE HEAT EXCHANGERS The next big category in geometrical designs is the so called plate-type design. Just as the shell-and-tube design, the plate-type heat exchangers can be classified as prime surface heat exchangers. Plate heat exchangers retrieved their name from their geometrical construction in the form of plates. They usually consist out of several plates, with cut cavities that form flow channels when stacked together with other plates . The number of plates can be varied in order to adjust the internal heat transfer surface area. The plates are usually sealed by gaskets, welded together or brazed. A scheme of a gasketed plate heat exchanger can be seen in figure 2.6. Advantages and disadvantages of plate heat exchangers are the following: 2.1.H EAT EXCHANGER 15

• Very Good cleaning capabilities (gasketed type)

• High ﬂexibility (gasketed type)

• Good fouling resistance (10-25% of shell-and-tube designs)

• High heat transfer rate

• Very high thermal efﬁciency

• Compact design

• Generally cannot withstand high pressures

• Generally cannot withstand high temperatures

• Generally cannot withstand high pressure or temperature differences

• High pressure drop for equivalent ﬂow velocities

Figure 2.6: Gasketed plate and frame heat exchanger [4]

The plate type heat exchangers can be sub-divided into the gasketed-, the welded-, the spiral plate-, the lamella-, the printed-circuit- and the panelcoil-heat exchanger. Their differences will be outlined briefly. The main characteristic of a gasketed heat exchanger is, that the individual plates are sealed by gaskets. This limits the pressures and temperatures possible to the gasket features. The plates are overall compressed by a bolt system and commonly fabricated out of stainless steel or titanium. The stamping of the individual plates has a major influence on the flow pattern within the heat exchanger. Due to this there is a variety of stamping patterns available for this type of heat exchanger. It is however difficult to obtain a precise flow arrangement like counterflow or parallelflow, as it is obtainable e.g. by double-pipe exchangers.

The build up of the welded type plate heat exchanger is in principle the same as for the gasketed one. How- ever higher operating pressures and temperatures can be achieved because the individual plates are welded together. This improves the temperature and pressure usability, however also limits the usability in terms of ﬂexibility and cleaning.

The spiral plate heat exchanger consists usually out of sheet metal being spirally wound and welded to two end plates. They usually occupy a relatively large volume and for non corrugated sheet metal the heat transfer coefficient is less than e.g. for a gasketed plate type, but larger than for a shell-and-tube type heat exchanger. The spiral type heat exchanger can accommodate fluids with a high solid content or high viscosities due to its high fouling resistance and ease of cleaning. It however allows fluids only to have low pressures and temperatures compared to the shell-and-tube type. 16 2.L ITERATURE SURVEY

Lamella heat exchangers are closely related to a tubular design since they feature several lamellas in a shell instead of an internal tube bundle. This exchanger type however uses gaskets to prevent leakage and therefore suffers from the same pressure and temperature limitations as the gasketed plate type heat exchanger.

The next heat exchanger type is the so-called Printed-circuit heat exchanger. This heat exchanger is fabricated similarly to printed circuit boards. The plates featuring etched channels are then combined by diffusion bonding. The in- and outlets are usually welded. As for the gasketed type heat exchanger the ﬂow pattern is dependent on the layout of the channels on the plate. This type of heat exchanger features a high rate of compactness and allows operating pressures up to 50 MPa at temperatures up to 800◦C. This, added to the ﬂexibility is the main advantage of this heat exchanger design.

The Panel-coil heat exchanger is composed out of multiple panels forming a channel inside them. They are commonly used as drop-ins for tanks or exchange heat with surrounding air. The panels are custom made and are adapted to the required size by adapting their width height and number of units. They are manufactured by point welding two plates with each other and then push them apart by applying pressure on them. Hence the welding points form the position edges, while the remaining surface is separated apart, forming channels. These panels are a very cost effective solution but are limited to very low pressures.

EXTENDEDSURFACEHEATEXCHANGERS

In contrast to above presented heat exchangers, extended surface heat exchangers can be of various forms due to the uniqueness of the ﬁns applied. Their main difference to the above mentioned forms of heat exchangers is, that they make use of ﬁns to increase the heat transfer surface above the possible surface created by a prime surface heat exchanger arrangement. By doing so very high values of heat exchanger effectivenesses can be obtained. This good feature can be combined with some design features of other kinds of prime surface heat exchangers. The result is often a cross-over between two types of heat exchangers. Their strengths and weaknesses are therefore dependent on what type of heat exchanger they are combined with. However in general they feature:

• Very high effectivenesses possible

• High ratio of compactness

• Flexible and combinable design with prime surface heat exchangers

• Very high surface area per unit volume possible

• Very high pressure differences possible

• Temperature limited only by material and bonding

• Cleaning and fouling can become a problem

• Unique designs are usually very cost intensive

This analysis makes clear, that extended type heat exchangers are preferred when it comes to designs involv- ing fluids with big differences in heat transfer capability, such as liquids and gases. Because of the manifold design opportunities in extended surface exchangers, two very common types will be presented, namely the Plate-fin Heat Exchanger (PFHE) and the tube-fin heat exchanger, as they are also presented in [4].

The first design to be discussed is the plate-fin heat exchanger. This type of exchanger is made out of corrugated fins being bound together by two plates. The plates also serve as a boundary for the two fluids passing through. A well-known example is a car radiator, as depicted in figure 2.7. The fins used can have multiple shapes to improve the heat transfer depending on the flow medium and the requirements. While most of the designs operate in an area of up to 8.3 MPa, some examples show pressure capabilities of more than 35 MPa. The material choice for the plate-fin type of heat exchanger is also very manifold. Depending on the operating pressures and temperatures materials for plates, fins and for bonding have to be selected. Those are usually metals, but also ceramics were already used in very high temperature applications above 1000◦C. In contrast to this design, Tube-fin heat exchangers are more related to a tubular design than the before mentioned layout. They feature mostly a tube, around which fins are oriented in different ways, depending on the flow direction. Two examples can be found in figure 2.8. They are often used in condensing applications 2.1.H EAT EXCHANGER 17

Figure 2.7: Fin automotive condenser [4] and application with a huge pressure difference. The temperatures achievable are limited by the bonding of the ﬁns and the materials used.

Figure 2.8: a) individual finned tubes b) flat continuous fins on an array of tubes [4]

2.1.5. MECHANICAL ASPECTS The mechanical features of the heat exchanger are dependent on the geometry of the heat exchanger. In principal the requirements on the heat exchanger are similar to those for the geometry, but some have to be added. The operating environment puts the following constraints on the Heat Exchanger (HEx):

• High Temperature

• High Pressure

• High Temperature difference for the ﬂuid-ﬂuid boundary material

• High Pressure difference for the ﬂuid-ﬂuid boundary material 18 2.L ITERATURE SURVEY

• Fabricability • Fatigue resistance • Corrosion resistance • Mountability When a design is put into practice many other factors have to be taken into account. As already indicated a mechanical analysis is not made in depth. An analysis regarding thermal stresses, long time operation and resistance to Foreign Object Damage (FOD) is not performed. Taking a tubular design as likely, the material thickness requirements dictated by the operating pressure and temperature can be determined. In order to do so it is needed to have access to a material database in order to determine the yield stresses of different materials as function of temperature. With this known there are two basic types of stresses. The ﬁrst one is the hoop stress. The hoop stress gives the circumferential stress due to the pressure within the vessel. By making use of formula 2.20, the required thickness of a cylinder can be determined as a function of the applied pressure and the maximum allowable stress. To calculate the necessary thickness of a sphere, equation 2.21 can be used. For these calculations a thin walled assumption was made [26].

Pr t (2.20) = σ Pr t (2.21) = 2σ In these equations t gives the thickness as a function of the pressure P and the radius of the shell r . σ stands for the maximum allowable stress. The second type of stress is axial stress. It can be computed in a similar manner. Also for this calculation the thin wall approximation was made. The respective formulae can be found in relation 2.22 for a cylinder. Pr t (2.22) = 2σ

2.1.6. SINGLE PHASE FLOW When flowing through the exchanger LNG will enter as a single phase fluid in a liquid state. Eventually it will start evaporating. This flow regime is called two-phase flow. After this process is completed, LNG will be present in gaseous form as a single phase fluid again. The air stream will always remain in a single phase (gas). This section will elaborate on the calculations necessary to determine the heat transfer and pressure drop for a single phase fluid. For the heat exchanger design it is assumed, that the flow is fully developed. According to Subramanian [27] this is a valid assumption for L 50. This condition might not be met by the d ≥ shell flow, but very likely by the tube flow. Since however the thermal entry region is even shorter than the hydraulic entrance length, it is assumed, that also here fully developed flow conditions apply.

HEAT TRANSFER For the single phase heat transfer multiple possibilities are given. The relations given below hold for idealized flow situations in standard heat exchangers. The influence of geometry on the flow behavior can however lead to a distortion of the flow, which can significantly impact the heat transfer. These effects cannot be modeled with the method given. Again a distinction has to be made for different Reynolds numbers. For laminar flow the Nu is a constant and related to the geometry as depicted in reference [28]. In order to determine the heat transfer the Nusselt number has to be found. Two correlations for the Nusselt number are depicted in relations 2.23 and 2.24. They will be used for Reynolds numbers below 3000.

Nu 4.364 (2.23) = Nu 3.66 (2.24) = Equation 2.23 will be used for constant heat flux, while formula 2.24 is being used for constant wall temperature. Since this method only holds for laminar flow, a further model is needed to be able to predict heat 2.1.H EAT EXCHANGER 19 transfer in turbulent flow. For turbulent flow multiple models are known. For this master thesis the popular Gnielinski model will be used to determine the single phase Nusselt number. The correlation was obtained by [28] and can be applied for multiple flow geometries:

fD /8(Re 1000)Pr Nutur − (2.25) = 1 12.7(f /8)0.5(Pr 2/3 1) + D − The friction factor is deﬁned by the Filonenko relation, given in formula 2.26, which is valid for Reynolds numbers above 3000 and thus also for the range this equation is valid.

2 f (0.79ln(Re) 1.64)− (2.26) D = − The equations presented will be used for both, the liquid as well as for the gaseous phases. If fins are added to tubes or plates, the heat transfer will alter since more heat can be absorbed or released. Especially the unilateral increase in heat transfer surface area makes it valuable if the difference in heat transfer coefficient between the two fluids becomes large. The discussion is based on reference [4] and discusses rectangular fins with constant cross section. From [4] it emerges, that the temperature profile and the resulting heat transfer coefficient can be obtained by relation 2.27 2 d θ d(ln Ak x ) dθ m2θ 0 (2.27) dx2 + dx dx − = with θ(x) T (x) T (2.28) = − ∞ and hP m2 (2.29) = k f Ak x When the assumption of a thin, straight fin of rectangular cross section is made relation 2.27 can be reduced to: d 2θ m2θ 0 (2.30) dx2 − = Formula 2.30 is a differential equation to which the solution is of the form

mx mx θ C e− C e (2.31) = 1 + 2

The constants C 1 and C 2 have to be evaluated according to the boundary conditions. For the adiabatic tip ﬁn which will be used throughout this thesis, the boundary conditions are as given in relations 2.32 and 2.33. The ﬁrst one indicates that there is heat transfer through the tip is zero, while the second one assumes that the base temperature is equal to the wall temperature.

µ dθ ¶ q x l k f in Ak 0 (2.32) = = − dx x l = = θ(0) T T θ (2.33) = − f ree = 0 Based on this equation the temperature distribution can be found for the fins. Making the assumption that all the heat entering the fin is also leaving the fin and thus being transferred to the other fluid implies a relation given in equation 2.34. q q q q (2.34) conv = 0 − l = 0 Therefore the term of q0 has to be found in order to determine the heat flux through the fin. By solving equations 2.27 to 2.31 equation 2.35 can be obtained.

hP cosh(2mx) 1 q0 θ0 − (2.35) = m sinh(2mx)

Adjusting the total fin heat transfer surface area according to the number of fins used will result in the additional heat flux by the fin. 20 2.L ITERATURE SURVEY

PRESSUREDROP

The single phase pressure drop of a heat exchanger consists out of many contributions. A detailed discussion is given by Shah [4], which will be presented here. The complete pressure drop can be resembled by equation 2.36. X ∆p ∆p ∆p ∆p ∆p (2.36) tot = i + e + core + bend The total pressure drop therefore is the sum out of the single pressure drops. Those are:

• Pressure drop due to irreversible effects at the entrance of the heat exchanger

• Pressure rise due to irreversible effects at the exit of the heat exchanger

• Pressure drop due to the core pressure drop

• Pressure drops due to bends

The core pressure drop again can be divided in two parts. The first part stems from the friction within the core of the heat exchanger, while the second part origins the change in density and thus represents a momentum pressure drop. According to Shah [4], the entrance pressure drop consists out of two phenomena. The first contribution is a pressure drop due to the change in cross-sectional area when the fluid enters the exchanger. The second part comes from the irreversible expansion which follows an abrupt contraction.

The formulae to calculate the entrance effect can be retrieved from relations 2.37 and 2.38

2 G ¡ 2¢ ∆p A 1 σ (2.37) = 2gc ρi −

G2 ∆p Kc (2.38) = 2gc ρi

While the factor of σ is deﬁned in relation 2.39, Kc is an empirical measure. It is dependent on σ as well as on the Reynolds number and the geometry. Together with Ke it can be retrieved from ﬁgure 2.9. In a similar

Figure 2.9: The factor Kc and Ke as a function of σ [4] manner, the exit effect of the exchanger is built up on a change in cross-sectional area of the exchanger and an 2.1.H EAT EXCHANGER 21 irreversible free expansion with accompanying momentum rate change. The respective formulae are given in relations 2.40 and 2.41. Ao,2 Ao,3 σ (2.39) = Ao,1 = Ao,4 2 G ¡ 2¢ ∆p A 1 σ (2.40) = 2gc ρo − G2 ∆p Kc (2.41) = 2gc ρo Depending on the number of changes in cross section within a heat exchanger, theses formulae have to be used multiple times, not only once at the entrance and exit of the heat exchanger. This can be of importance to the exchanger when the fluid is guided into the exchanger. The fluid then undergoes two cross-sectional area changes. The first one happening when the fluid is entering into the heat exchanger head and the second one when the fluid enters the core. The same holds for the heat exchanger exit. Within the core also two effects are given. The first one being due to the change in momentum. It is represented as the change in density over the exchanger and can be found in relation 2.42.

G2 ∆pmom (νo νi ) (2.42) = gc −

In this relation ν gives the respective specific volume at the entrance and exit of the section. The other contribution from the core pressure drop comes from the frictional pressure drop. This pressure drop is usually calculated from a friction factor and an according relation to the pressure loss. The friction factor is dependent on Reynolds number and the surface roughness of the fluid channel. The pressure loss equation for single phase flow can be retrieved from equation 2.43.

G2 ∆p f fF L νm (2.43) = 2gc rh · ·

In this equation rh gives the hydraulic radius of the geometry, and fF the friction factor. νm gives the average specific volume of the section. The friction factor is different for laminar and turbulent flow regimes. For laminar flow, relation 2.44 is given by Barron[3], which is valid for Re 2300. For turbulent flow a more recent < developed relation can be used. An equation featuring great accuracy and wide applicability according to Fang [29] is given by Fang and Xu [30] and is depicted in equation 2.45. The validity range is 3000 Re < < 108. 64 fD 4 fF (2.44) = · = Re · µ ¶¸ 2 150.39 152.66 − fD 0.25 log (2.45) = Re0.98865 − Re Between the two relations an interpolation can be made to ensure continuity. Since the validity range of the Reynolds numbers is not very far apart, the introduced error is estimated to be negligible. The hydraulic diameter plays a major role in this correlation since it involves the wetted perimeter. This quantity is important for the friction factor. The complete correlation of the pressure loss along a single phase heat exchanger according to Shah then becomes:

2 · µ ¶ µ ¶ µ ¶¸ G 2 ρi L 1 ¡ 2 ¢ ρi ∆ptot 1 σ Kc 2 1 fF ρi 1 σ Ke (2.46) = 2gc ρi − + + ρo − + rh ρm − − − ρo

Depending on the layout of the heat exchanger there is a need to incorporate bends in the design. For the given design this can be of importance since the volume available for the heat exchanger is limited. Since bends come with an increase in friction on the outside bend due to an increase in dynamic pressure as well as a decrease in friction on the inside of the bend. Since the two effects do not cancel there is a net effect in pressure loss. This effect is a.o. dependent on the curvature of the bend, the radius, the diameter of the tube and the Reynolds number. In general the bend effect is represented in equation 2.47.

2 ρum ∆pbend Kb,t (2.47) = 2gc 22 2.L ITERATURE SURVEY

The values of the velocity u and the density ρ have to be evaluated at the entrance of the bend. The factor Kb,t is the total pressure drop coefﬁcient of the bend and is of semi-empirical nature. It is built up as depicted in relation 2.48. 4L Kb,t Kb K f Kb fF (2.48) = + = + Dh

In this relation Kb represents the pressure drop due to the curvature effect, the ﬂow development effect and the roughness of the tube. The factor K f resembles the pressure drop associated with the outlet of the bend into a straight pipe.In this relation fF is calculated by the fanning friction factor. The calculation of Kb is of semi-empirical manner and can be retrieved from relation 2.49.

K K ∗C C C (2.49) b = b Re dev r ough This formula consists out of the pressure drop coefficient due to the bend evaluated at a Reynolds number of Re 106, a correction factor C to account for the actual flow Reynolds number, a correction factor C to = Re dev incorporate the flow development in the outlet of the bend and a factor incorporating the surface roughness Cr ough. The last one can be calculated according to equation 2.50, with fF,smooth being the friction factor for a hydraulically smooth tube and fF,r ough the equivalent for the roughness assumed for the actual tube.

fF,r ough Cr ough (2.50) = fF,smooth

Values for the friction factors can be obtained from ﬁgure 2.10.

Figure 2.10: Rough and smooth friction factors for equation 2.50 [4]

The factor of Kb∗ can be retrieved from figure 2.11. The factors of CRe and Cdev are obtained differently r c depending on their value. The factor of CRe shows strong dependence on the Reynolds number for 1. di < r c It has to be calculated as indicated in the following. For 0.7 1 or Kb∗ 0.4 the factor of CRe should < di < < be directly determined by figure 2.12 for a ratio of r c 1. Else it has to be calculated according to formula di = 2.51. Kb∗ CRe (2.51) = K 0.2C 0.2 b∗ − Re 0 + r c In this relation the factor of CRe 0 is obtained from figure 2.12 for a ratio of 1. The parameter of Cdev is di = r c obtained from figure 2.13. If 3 or the bend is larger than 100°, then Cdev 1. With the large amount of de- di > = pendency on graphical empirical values it will not be possible to implement the bends in the heat exchanger program. Those will therefore have to be added manually when the design is finished. 2.1.H EAT EXCHANGER 23

Figure 2.11: Kb ∗ as a function of the bend, curvature and tube diameter [4]

2.1.7. MULTIPHASE FLOW

The calculation procedure of the phase change of LNG differs strongly from the one of a single phase fluid. This holds for both, pressure drop as well as heat transfer. This comes mainly from the fact that the fluid behavior is not as linear anymore as it is for a single phase fluid. The multiphase fluid behaves like a mixture between a gas and a liquid with a character changing depending on the vapor quality of the fluid. The effects have an influence on the heat transfer as well as the pressure drop.

HEAT TRANSFER

The process of two-phase heat transfer is very complex. For this reason, like for the pressure drop, empirical methods have to be applied. The heat transfer is assumed to be independent of the pressure, a valid assumption according to Chen[31], who states, that with increasing inlet pressure the effects on heat transfer are weak. Since before and after the two phase part there is single phase ﬂow, it is not necessary to implement entrance and exit effects for the two phase part. Like for the single phase pressure drop, there are many mod-

Figure 2.12: CRe as a function of Reynolds number [4] 24 2.L ITERATURE SURVEY

Figure 2.13: Cdev as a function of Length over diameter and Kb ∗ [4] els available for calculating the two phase heat transfer [32]. A method already chosen for a LNG application by Chen[31] is the Liu-Winterton relation, presented in the paper of Zou[33]. The Liu-Winterton relation will therefore be presented in the following.

As many authors cite, the two phase heat transfer is mainly dependent on nucleate boiling and forced convection boiling. A figure of the different boiling phenomena can be retrieved from figure 2.14. As it can be seen the heat transfer rates increase strongly at the nucleate boiling part of the boiling curve. This part climaxes in the Departure from Nucleate Boiling (DNB). The Critical Heat Flux (CHF) is obtained at this point. When the superheat temperature is further increased, excessive bubble growth prevents new liquid mass to flow to the wall. This leads to a significant increase in wall temperature and a decrease in heat transfer coefficient. The point of CHF is therefore to be avoided, since it can lead to failure of the heat exchanger. To date however the point of CHF very difficult to predict and only few methods attempt to do so. For this reason evaluation of the CHF will not be further pursued. In order to be able to solve the problem, the wall temperature must be found

Figure 2.14: Boiling phenomena and associated qualitative heat ﬂuxes [6] 2.1.H EAT EXCHANGER 25

first. As only the bulk fluid temperatures are given, this is difficult to achieve as heat flux and heat transfer coefficient both depend on the wall temperature in the two phase region. For this reason the assumption was made that the wall temperature can be calculated as depicted in [34]. In this paper by Sparrow the inside bulk fluid temperature was determined by measurements of a thermocouple mounted to the outside of the tube. The simple equation for prediction of the bulk temperature was rewritten to obtain the wall temperature from the bulk fluid.

At ﬁrst, a critical measurement point value θ was calculated. This can be found in relation 2.52. The wall temperature then can be found by inserting the found value of θ as a function of the ﬂuid Prandtl number into equation 2.53. θ 0.784Pr 0.0358 (2.52) = T θT T (1 θ) (2.53) w = LNG + Air − The values of validity from the paper can be found for by the maximum and minimum values for the physical parameters. Those areas are depicted in table 2.5

Table 2.5: Validity range for wall temperature calculation

Limit Reynolds number air Reynolds number ﬂuid Prandtl number ﬂuid Min 5,000 10,000 1 Max 30,000 150,000 50

The fluids used in the given paper are water and oil respectively. A major problem can be seen in the calculation of the heat flux. Since only the bulk fluid temperatures are known the heat flux must be determined first. This is also given in the article of Liu and Winterton [35]. At first the superheat temperature has to be known. It is defined in relation 2.54 and physically stands for the temperature difference between the saturation temperature of a fluid and the hotter wall temperature. It therefore gives the temperature to which the wall is hotter than the saturation temperature of the fluid. ∆Tsu has to be a positive quantity in order to initiate boiling. ∆T T T (2.54) su = w − s With this known the and the help of relations 2.61 and 2.65, equation 2.55 can be written.

2 q2 (F h ∆T )2 ¡A S∆T ¢ q4/3 (2.55) = L su + p su with Ap being deﬁned as in relation 2.56.

0.12¡ ¢ 0.55 0.5 A 55p log p − M − (2.56) p = r − 10 r By deﬁning equations 2.57, 2.58 and 2.59, the general form as depicted in relation 2.60 can be obtained.

q F h ∆T (2.57) L = L su · q ¸2 · q ¸2 q 3 (2.58) ∗ = F hL∆Tsu = qL

µ ¶2 Ap S 4/3 C qL (2.59) = F hL

q 3 C q 2 1 0 (2.60) ∗ − ∗ − = This is a cubic equation that can be solved for q. According to Liu and Winterton[35], the equation has only one non imaginary root, which gives the heat flux. With this heat flux known, the heat transfer coefficient can be calculated. The same holds for the amount of LNG that can be vaporized by this amount of energy input. 26 2.L ITERATURE SURVEY

Chen[36] was the ﬁrst to implement the heat transfer in two phase ﬂow as a superposition between a forced convection term and a nucleate boiling term. The Liu-Winterton relation will be based on both of them. The general correlation is depicted in equation 2.61.

q 2 h ¡F h ¢ (S h )2 (2.61) tp = · sp + · nb Where the subscripts stand for tp=two-phase, sp=single phase and nb=nucleate boiling. The individual components are then presented in equations 2.62 to 2.64.

0.8 0.4 kt,l hsp 0.023(Rel) (Prl ) (2.62) = Dh

0.35 F £1 xPr˙ ¡ρ /ρ 1¢¤ (2.63) = + l l v − Gd Rel (2.64) = µl

Equation 2.62 gives the well-known Dittus-Boelter relation. x˙ represents the average vapor quality. F hsp resembles the contribution of the forced convection, while Shnb gives the part of nucleate boiling. hnb can be determined from relation 2.65, from [37].

0.12¡ ¢ 0.55 0.5 0.67 h 55p log(p ) − M − q (2.65) nb = r − r The factor S is the suppression factor and can be determined from relation 2.66.

0.1 0.16 1 S (1 0.055F Re )− (2.66) = + l

In relation 2.65 M gives the molecular weight and pr the reduced pressure, which is the actual pressure divided by its critical pressure. The critical pressure of LNG is according to reference [38] roughly 4.64 MPa. For the reduced set of constituents the data obtained from source [23] however suggest a value of 5.23 MPa. Since the NIST database is the primary data source for this master thesis, also the value for the critical properties will be adopted from it.

PRESSUREDROP Currently there are numerous methods for calculating the two phase pressure drop. In most cases the two phase pressure drop is an effect that can be split into two sub effects. The first one being the effect of friction due to shear stress the fluid builds up when an interaction with a solid boundary is given. The second effect comes with the change in momentum of the fluid. Because the gaseous phase has a lower density it will flow at a higher speed through the channel. This velocity difference results in an acceleration of the liquid phase by the gas phase and in a deceleration of the gas by the liquid. The total two phase pressure drop therefore can be calculated as the sum of the two individual components as depicted in relation 2.67.

∆p ∆p ∆p (2.67) tot,tp = a + f The pressure drop due to acceleration of the ﬂuid will be modeled according to Mudawar [39] and is depicted in relations 2.68 and 2.69.

(" 2 2 # " 2 2 #) 2 vg xe,out v f (1 xe,out ) vg xe,in v f (1 xe,in) ∆pa G − − (2.68) = α + (1 α ) − α + (1 α ) out − out in − in The calculation of the void fraction will be done according to Zivi [40] as suggested by Mudawar [39].

" µ ¶µ ¶2/3# 1 1 xe v f − α 1 − (2.69) = + xe vg

In the paper of Fang [29] an evaluation of the mostly used methods for determining the frictional two-phase pressure drop is given. Among the best ones are the Müller-Steinhagen and Heck model as well as the Friedel model[29]. The author finally gives a new method to model the two-phase pressure drop. Because however 2.2.T URBINECOOLING 27 most of the used experimental data comes mostly from refrigerants it cannot be concluded, that the presented relation is also valid for the given research with LNG flow. Because the Müller-Steinhagen and Heck model was among the best predictive models over a large range of flow regimes and because it was given by Chen [41] as a method that can proved acceptable solutions for LNG, it is adopted and presented here. Be- cause the list of further separated flow models and homogeneous flow models is very long, it is not possible to outline all of them here. In the discussion presented in the following the Müller-Steinhagen and Heck model will therefore be shown representatively for the separated flow models. Homogeneous flow models treat the fluid in general as a mixture between its vapor and liquid phase. They won’t be pursued further. The calculation method for the frictional pressure drop in two phase flows is therefore adopted from Müller- Steinhagen and Heck [19]. The equations necessary to express the frictional pressure loss in two-phase flow are given in relations 2.70 to 2.74. µ dp ¶ m˙ 2 ζl A (2.70) dL f ,l = 2ρl d = µ dp ¶ m˙ 2 ζv B (2.71) dL f ,v = 2ρv d = Müller-Steinhagen and Heck approximate the first 70% of the phase change by a linear combination of A and B, found in relation 2.72. For the complete range of vapor quality, the relation adds the terms of a single phase vapor for the last 30 % and superimposes the two terms. The result can be found in relation 2.73. The complete equation of the Müller-Steinhagen and Heck method can be seen in 2.74.

G A 2(B A)x˙ (2.72) = + − µ dp ¶ G(1 x˙)1/3 Bx˙3 (2.73) dL f ,tp = − + µ ¶ · 2 µ 2 2 ¶ ¸ µ 2 ¶ dp m˙ m˙ m˙ 1/3 m˙ 3 ζl 2 ζv ζl x˙ (1 x˙) ζv x˙ (2.74) dL f ,tp = 2ρl d + 2ρv d − 2ρl d − + 2ρv d

In these equations x˙ gives the quality of the two phase fluid. The friction factors ζl and ζv are dependent on the phase of flow since they resemble the frictional behavior of the respective phase. Depending on the phase and the Reynold’s number, the friction factor can be determined as follows: 64 64 • ζl and ζv for Rev,Rel 1187 = Rel = Rev ≤ 0.3164 0.3164 • ζl 1/4 and ζv 1/4 for Rev,Rel 1187 = Rel = Rev > md˙ md˙ • Rel and Rev = µl = µv With these terms, the frictional pressure drop in the two phase flow can be modeled. This completes the discussion on the two phase pressure drop.

2.2. TURBINECOOLING

The reason why the heat exchanger is needed is to cool down the turbine in a more efficient manner. Because this alternative is given due to the cold reservoir of cryogenic fuel, a benefit in thrust specific fuel consumption should be gained due to the higher Turbine Inlet Temperature (TIT)s possible. On the other hand it would be possible due to the better cooling of the turbine, to reduce the bleed air mass flow. Ultimately the cooler turbine is less subjected to wear and therefore the maintenance interval for the turbine could be increased. For gas turbines multiple ways of cooling are available. This variety of cooling techniques arose from a need for high cooling capabilities. In figure 2.15 a development of cooling in terms of allowable TIT is given. As it can be seen, from the early days of gas turbines to nowadays the TIT has been increased steadily. From a certain point on, the thermal capabilities of the materials alone were insufficient to bear even higher TIT’s. At this point cooling was applied for turbines. These achievements were made due to developments in material as well as cooling techniques. In the following, these techniques will be briefly elaborated on. In total three relevant cooling techniques for blade cooling will be discussed: 28 2.L ITERATURE SURVEY

Figure 2.15: Turbine inlet temperature increase over the decades [7]

1. Convection cooling

2. Jet impingement

3. Film cooling

The first two techniques are internal blade cooling arrangements, while film cooling is an externally applied cooling. Besides the heat transfer, pressure drop of these cooling methods will be addressed. The pressure drop originating from active turbine cooling gives the second major component in pressure drop (the first one being mentioned above from the heat exchanger). Following this definition the total allowable pressure loss consists out of the heat exchanger loss, the piping loss and the turbine blade cooling loss. This pressure loss in total is not allowed to be higher than the pressure loss that can be found in the combustion chamber 1 (for combustion chamber 2 there exists no real limit, because after expanding the flue gas in HPT, the pressure drops significantly). Therefore an estimation of the pressure drop over the turbine has to be performed. The pressure drop is dependent on the cooling method applied, the geometry and composition of the channel through which the air is flowing as well as on the fluid parameters. Since turbine cooling is not explicitly dealt with in this thesis but adopted from the tool provided by Tiemstra [11], it will be discussed only briefly.

2.2.1. CONVECTION COOLING

Convection cooling is the oldest form of active cooling and present in almost any modern gas turbine. The principle itself is very simple: cold bleed air from the high pressure compressor is led through a channel. The heat of the hot gas stream enters the turbine blade mainly over forced convection. In the blade itself conduction transports the heat to the cooling channel. Heat is then removed from the blade by forced convection and via the heated cooling air led back into the core stream. The adaptations within an engine in order to be able to guide a cooling gas stream from the High Pressure Compressor (HPC) to the HPT are mechanically significant. Also the increase in t the design of a turbine blade is huge. In figure 2.16 a cut through a convective cooling turbine blade is given. The geometry of the convection channels makes the calculation of heat transfer and pressure drop difficult. It comes on top, that nearly every gas turbine features another layout in terms of cooling channeling. Furthermore many secondary effects taking place within the cooling channel add uncertainty to the problem. 2.2.T URBINECOOLING 29

Figure 2.16: Cut through a convective cooling turbine blade [8]

HEAT TRANSFER To calculate the heat transfer of a cooling channel, Newton’s law of cooling can be applied. It is given in 2.75. q h A(T T ) (2.75) conv = w − cool In modern engines this cooling technique is frequently applied. The performance of such cooling systems is even enhanced by the application of ribs in the channel to foster turbulence (figure 2.17 b)). This is good on the one hand because it increases the maximum heat transfer possible, on the other hand however increases the pressure loss and contributes to an increase in complexity. Ribs can be put into a cooling channel in different shapes and layouts. The improvement of cooling is dependent on this shape as well as on the flow Reynolds number. The benefit of ribs comes mainly from the refreshment in boundary layer after every rib and an increase in turbulence. A second enhancement method possible is the application of pin fins. An example of pin fins can be seen in figure 2.17 b). Pin fins stretch over the entire flow channel height and can be arranged in line or in a staggered way. They benefit from the effect, that the internal cooling area is increased and the formation of a fresh new boundary layer, which takes place on each pin fin. Pin fins are usually employed at the trailing edge channel of the blade and exist in various shapes.

Figure 2.17: Cooling enhancement by a) ribs and b)pin ﬁns [9]

PRESSUREDROP For convection cooling it is assumed, that the internal channels are ideal and hence can be evaluated according to relation 2.43 from the heat exchanger discussion. It is assumed, that the internal ﬂow in the channel is turbulent. Since the pressure drop in a single phase ﬂow is only dependent on the geometry of the channel, 30 2.L ITERATURE SURVEY

Table 2.6: Coefﬁcients for Nusselt number determination from Florschuetz(1981) [15]

In line Pattern Staggered Pattern

C nx ny Nz C nx ny Nz A 1.18 -0.944 -0.642 0.169 1.87 -0.771 -0.999 -0.257 B 0.437 -0.095 -0.219 0.275 1.03 -0.243 -0.307 0.059 m 0.612 0.059 0.032 -0.022 0.571 0.028 0.092 0.039 n 0.092 -0.005 0.599 1.04 0.442 0.098 -0.003 0.304

Reynolds number, Prandtl number and surface roughness, the calculation of one blade per stage is sufficient to determine the pressure drop of an entire stage. Additional attention however should be spent, if bends,T- sections, ribs or pin fins occur in a flow. Those impede the flow and hence contribute further to the pressure drop.

2.2.2. JETIMPINGEMENT Even though jet impingement is also a convective cooling technique, it is an active cooling method coming with different heat and pressure behavior. In jet impingement, air flows from a region of higher pressure through a hole to a region of lower pressure by forming a jet. Mostly, designs feature an array of holes, that can be in line or staggered and that are close to the surface on which the jet impinges. The optimal distance between the jet holes and the impinging surface is dependent on the pressure delta, the number and spacing of the jets, cross flow and jet hole geometry. Like this a considerable cooling effect is achieved. The high cooling rates that can be achieved make jet impingement the primary choice for leading edge cooling and turbine vane cooling. A tribute to the high heat transfer coefficients is however also a higher pressure loss. Additionally jet impingement suffers from cross flow in an array and impose additional difficulties in manufacturing.

HEAT TRANSFER

The heat transfer coefﬁcient achievable with jet impingement can be calculated with formula 2.76.

Nu kt h · (2.76) = Dh

The Nusselt number itself can be determined as presented by the work of Florschuetz[15] and is given in relation 2.77. n Nu A Re m ©1 B£(z/d)¡G /G ¢¤ ªPr 1/3 (2.77) = · j − c j In this relation Gc /G j is the ratio of the cross ﬂow velocity over the jet velocity and (z/d) gives the dimensionless impingement height. The coefﬁcients A,B,m andn can be retrieved from table 2.6 in combination with equation 2.78. A,B,m,n C(x /d)nx (y /d)ny (z/d)nz (2.78) = n n An alternative representation for the heat transfer was given by Martin (1977), however because only descrip- tions of his method and not the original article could be retrieved, it was unclear, what the assumptions and range of validity of his method were, for which reason the one from Florschuetz will be shown.

PRESSUREDROP

Not many analytical relations exist to accurately model the pressure drop in a jet impingement array. This is due to the many influences of an array including entrance, exit jet impingement and cross flow effects. Since also the array only can be idealized with respect to the always different geometry in real life applications it is difficult to come up with a general relation. An approach for such a method was documented by Levy [42]. The total pressure drop according to his paper consists out of the pressure drop due to the following components: 2.2.T URBINECOOLING 31

• Inlet • Outlet • Friction inside the nozzle • Cross flow The inlet pressure drop coefficient can be calculated by relation 2.79. µ ¶ Ain ξn,in 0.5 1 (2.79) = − Ach,in The nozzle friction coefficient can be determined by equation 2.80. For exit conditions the pressure drop coefficient can directly be set to 1 assuming zero kinetic energy left, ξ 1. n,out = fD L ξ · (2.80) n,f r = D

In this relation fD gives the Darcy friction factor which has to be evaluated according to the flow type. The cross flow pressure drop can be evaluated directly by making use of relation 2.81: 1 ∆p ρV 2 (2.81) ch = 2 ch,out The nozzle specific pressure drops can be calculated as depicted by formula 2.82, while the entire pressure drop within a jet impingement array with cross flow can be calculated as the sum of the individual contributions as given in equation 2.83. µ 2 ¶µ ¶ X ρVi,n m˙ i ∆pn ξn (2.82) = 2 m˙ tot ∆p ∆p ∆p ∆p ∆p (2.83) tot = n,in + n,out + n,f r + ch,out

2.2.3. FILM COOLING Film cooling is the last active cooling technology presented. Film cooling is in contrast to the other cooling techniques no reactive, but a pre-active method. Instead of cooling down the material, heat is prevented to reach the blade material and thus forming a protective layer around the blade. When film cooling is applied, cold cooling air is ejected out of many holes in the surface of a turbine blade to form a protective film that is laid around the blade by the momentum of the air flow and the orientation of the jet holes. Film cooling is an external cooling measure and meant to increase the heat transfer resistance to the blade. It can be achieved by several (rows) of holes or by applying a porous material as blade material. While the first approach represents a discrete way of putting a film around the blade the second choice distributes the cooling film more evenly. While this approach prerequisites the use of a porous material, the approach of discrete holes has the advantage to put the cooling film where it is actually needed and therefore save cooling air. The design of the cooling holes is manifold. Holes change in diameter and shape as well as surface constitution. Moreover, in some gas turbines slots can be found instead of holes. Of great influence on the cooling and pressure drop the individual hole size and layout (angle, shape) as well the blowing ratio can be seen. The interplay between all these factors is important to produce a uniform and closed film over the surface to be cooled. The modeling of cooling performance when film cooling is applied represents a huge jump in complexity. This has multiple reasons: • Secondary Flow fields disturb cooling flow

• Determination of the ﬁlm temperature T f ilm • Large change in cooling performance and pressure drop by small geometry changes (as produced e.g. by a Thermal Barrier Coating (TBC)) • Failure of ﬁlm cooling can completely destroy the turbine • High levels of turbulence make computation very hard 32 2.L ITERATURE SURVEY

HEAT TRANSFER The heat transfer to the blade is also modeled in a different way. To determine the heat flux for a blade that does not use film cooling equation 2.84 can be used. With this calculation procedure there are clearly only two three involved, the free stream temperature, the wall temperature of the blade and the heat transfer coefficient. q h (T T ) (2.84) 0 = 0 f ree − w When film cooling comes into play and - only considering the local position a bit downstream of the injection - there are roughly three temperature layers involved, namely as before the hot free stream, then the cooling film and finally again the blade wall. This means, that relation 2.84 has to be changed into what is depicted in relation 2.85. q h(T T ) (2.85) = f ree − f ilm This makes the entire determination of the cooling very complicated because T f ilm is changing with its length and also with its height, due to the heat transfer processes involved in the turbulent mixing of the flows. This is illustrated in figure 2.18. The wavy blue arrows on the border of the cooling air flow to the free stream flow indicate the turbulent mixing and enhanced heat transfer. It can be seen from the color scheme, that the temperature of the film is not constant. The wavy black arrows indicate heat transfer. It has to be mentioned that this way of looking at film cooling is very much simplified. In reality turbulent mixing effects as well as no strict temperature layers play an important role in the cooling performance. For this reason nowadays still tests have to be performed to determine the overall cooling performance. In order to model the film cooling,

Figure 2.18: Film cooling - different heat zones [9] it is necessary to make significant assumptions. The major ones are: • Turbine blade shape is assumed to be a flat plate • No secondary flows • No leakages • Negation of turbulence enhanced flow mixing and cooling • Constant film temperature With these assumptions the cooling heat transfer coefficient can be determined in the following manner according to [43]. First the adiabatic wall temperature has to be determined. This can be done by relation 2.86. s c µ 2 3 p V Tw Ts · (2.86) = + kt 2cp The heat transfer coefficient of the cooling film cannot be determined directly. It has to be evaluated over the heat load ratio from relation 2.87 q h µ T f ree Tcool ¶ 1 η − (2.87) q = h − T T 0 0 f ree − w The term of η gives the cooling effectiveness and is defined as given in relation 2.88.

T f ree Tw η − (2.88) = T T f ree − cool 2.2.T URBINECOOLING 33

With the values of q and q0 being estimated at first, the heat transfer coefficient for film cooling can be determined. The estimation of q0 can be improved by resolving relation 2.89 and then combine it with the well known relation 2.76 Nu 0.0292 Pr 1/3 Re4/5 (2.89) = · ·

PRESSUREDROP

The pressure drop for film cooling is not calculated in the same manner as the one for convection cooling and jet impingement since the flow stream does not produce shear stresses with a boundary as this is the case for channel convection cooling or jet impingement. For film cooling the factor that says most about its pressure performance is the blowing ratio. It is defined in relation 2.90.

ρcVc M (2.90) = ρ V ∞ ∞ It gives the mass ﬂux of the jet with respect to that of the free stream. With the blowing ratio it can be steered how deep the stream of coolant will penetrate the free stream. When the blowing ratio is chosen too low, the high momentum of the free stream will prevent a proper protection ﬁlm from building up, while when it is too high the coolant stream shoots far in the free stream and exposes spots of the turbine wall to the hot temperature free stream. The blowing ratio is usually taken to be around M=0.5. This dictates a proper design for all engine thrust conditions at different altitudes.

2.2.4. BLADE MATERIAL The discussion of the loads applied is based on a presentation given by Tinga [10]. The material chosen for the blades is dependent on loads applied. While Nozzle Guide Vane (NGV)’s can theoretically be made out of ceramics because they do not move and are only exerted to temperature stress, rotor blades are made out of Cobalt- or Nickel alloys. This difference is needed in order to be able to cope with the loads due to rotation. The requirement on them are usually strength at a high temperature as well as creep, fatigue and oxidation resistance. Additionally the occurring vibrational loads also impose certain brittleness requirements on both, vane and rotor. In the past significant improvements in terms of the maximum allowable blade temperature and mechanical properties of turbine blades have been made. The biggest may origin from the reduction in total grain boundary length. While in the beginning of gas turbine production standard materials were used, it was later found out, that solidificated grain boundaries led to higher allowable loads. Nowadays turbine blades are manufactured out of a single crystal having no grain boundaries at all. Without any grain boundary the load applicable to a blade is close to the theoretical maximum. A comparison of the grain structure of a material can be seen in figure 2.19. As already mentioned, for the NGV’s theoretically ceramics can be used.

Figure 2.19: Material development a) polycrystal b) directionally solidiﬁed c) single crystal [10]

They are usually however not applied for aero-engines because of their brittleness. A further requirement for 34 2.L ITERATURE SURVEY the NGV’s is repairability and castability. NGV’s mainly have to be able to withstand corrosion, oxidation as well as creep and thermal fatigue. Additionally to the mentioned materials for the turbine blades the option for Oxide Dispersion Strengthened (ODS) alloys is given, providing very good high temperature properties. These hightech materials however are very expensive and therefore not always applied.

2.2.5. USED CODE

As outlined before, to calculate the turbine cooling use of the code of Tiemstra [11] will be made. The given code is based on analytical semi-empirical methods and can therefore not provide the details of more elaborated phenomena taking place. However for the given purpose of a preliminary design and the estimation of the coolant requirements for the heat exchanger the degree of accuracy is sufﬁcient. In this part, the code and the degree to which this code will be used will be explained.

THECODE

The code focuses mainly on turbine cooling. This part is dealt with very extensively and many ways of cooling can be represented in a very detailed fashion. The code allows to choose the type of cooling for a chosen rotor or NGV. For this stage, it can be chosen which cooling shall be applied. The choices channel cooling, jet impingement cooling, pin fin cooling and film cooling can be made. These cooling methods then offer further detailed freedom of choice in terms of number of channels, type of rib, number and layout of ribs, layout of holes for jet impingement etc. A snapshot of the GUI can be found in figure 2.20. The code also

Figure 2.20: Starting layout of the GUI of the code [11] comes along with a performance part, that can model the AHEAD engine under making several assumptions regarding parameters as pressure ratios, turbine inlet temperatures etc.

In order to evaluate the heat exchanger performance it is however most convenient, not to include all the available cooling features during the design. Hence only a standard stage will be produced, featuring one layout in terms of cooling. This stage can contain a detailed cooling arrangement, but is not allowed to change, since alternating cooling layouts will veil the performance of the heat exchanger module. For the development of an entire engine however, it makes sense to extend the code to as far as needed and to give a realistic view on the complete cooling of the turbine. This however is beyond the scope of this work and can be seen as recommendation for future work. 2.3.E NGINEPERFORMANCE 35

2.3. ENGINEPERFORMANCE

The engine performance will be dependent on the physical layout of the engine. In ﬁgure 2.21 the engine modeled will be shown As it can be seen, the engine consists out of the following performance relevant com-

Figure 2.21: Scheme of the hybrid engine [1] ponents: 1. Counter-rotating fan 2. Booster 3. High pressure compressor 4. Combustion chamber 1 5. High pressure turbine 6. Combustion Chamber 2 7. Low pressure turbine 8. Nozzle/Exit As final outcome, the effect of the heat exchanger should manifest itself in a reduced SFC and/or an increase in thrust. In the following it will be determined how the performance of the engine will be evaluated. Since not sufficient data for modeling the AHEAD engine was available at the time of coding, the data of the AHEAD’s baseline engine, the General Electric GE-90-94B was taken instead for the evaluation of the performance. Since the Lower Heating Value (LHV) of Kerosene and LNG is comparable, the assumption was made, that the GE-90-94B runs on LNG instead of Kerosene and the heat exchanger is applied to it. Additionally it should be mentioned that there is a difference in design points for an engine and for a heat exchanger. While the design point of an aero gas turbine is usually found at top of climb, the heat exchanger will have its design point at sea level conditions. For this the design point calculations shown in the following cannot be directly applied. Therefore use of GSP was made to evaluate the engine performance at off-design conditions. The calculations done by GSP deviate from the below shown calculation in several points, e.g. the physical properties like specific heat at constant pressure are not assumed to be constant, but varying with pressure and temperature. In general the procedure done by the tool however is the same. Its main advantage over the given calculations is however the possibility to derive an off-design performance estimation of the engine when the design point is given.

2.3.1. CALCULATIONS BEFORE ENGINE INTAKE The ﬁrst step is to determine the pressure and temperature at the engine inlet due to the ram effect of the engine. Using the ambient temperature and Mach number, equation 2.91 gives the temperature Tt,2, right before the fan. This temperature is used in connection to equation 2.92, to determine also the pressure at this stage. κair 1 2 Tt,2 T (1 − M ) (2.91) = amb · + 2 · 36 2.L ITERATURE SURVEY

κair µ ¶ κ 1 Tt,2 air − pt,2 pamb ( 1) ηi so 1 (2.92) = · Tamb − · + Next, the correct mass ﬂow at this altitude has to be determined. This is done using the given formulae 2.93,2.94 and 2.95. pamb δ (2.93) = 101325

Tamb θ (2.94) = 288.15

m˙ cor r δ m˙ · (2.95) = pθ With these calculations all parameters right in front of the air intake are known.

2.3.2. CALCULATIONS OF COMBINED AIRFLOW In this subsection the calculations of the flow parameters will be given before the flow is split up in core and bypass flow. The calculations are quite simple; the mass flow is the same as before and can therefore be overtaken from the previous subsection. The pressure undergoes an increase by the fan pressure ratio. This change may be calculated under the use of equations 2.96. The accompanying change in temperature can be determined by using equation 2.97. p p PR (2.96) t,21 = t,2 · f an  κ 1  µ ¶( air − ) Tt,21 1 pt,21 κair 1  1 (2.97) Tt,2 = + ηi so pt,2 −

In equation 2.97 use is made of the isentropic efficiency of the fan as well. After the fan the air flow is split in one core and one bypass flow. Because the bypass ratio is known as well as the mass flow before the fan, the air masses that go through the core and the ones through the bypass can be easily determined.

2.3.3. CALCULATIONS OF CORE GAS FLOW The biggest part of the engine cycle calculation is the core calculation. This flow calculation can be split up into multiple parts. First the air is compressed twice, then LNG is injected and burned in combustion chamber 1. The air is then expanded in the high pressure turbine and bio-fuel is injected in the second combustion chamber. The air is then expanded a second time over a low pressure turbine. The calculations for the booster are done in similar way as the calculations for the fan. The pressure ratio of both booster and compressor are given and with equation 2.97 the temperature at each point can be determined. The mass flow right before entering the combustion chamber is the same as right before the booster minus a small portion taken away as bleed air to cool the turbines. When entering the combustion chamber fuel gets injected and the hot air-fuel mixture gets ignited. To calculate the fuel mass flow, use of equation 2.98 can be made. m˙ 3 cp,g as (Tt,31 Tt,3) m˙ f · · − (2.98) = LCV η · cc The total mass flow is now the combination of the hot air minus the bleed plus the injected fuel. The fuel comes already at an elevated temperature. The positive influence of the heated fuel on the complete gas stream however is estimated to be small. It could be determined by the mixed masses of fuel and air and their combined temperature. This temperature could then replace Tt,31. Furthermore for the core cp,g as and κg as have to be used. The pressure drop over the combustion chamber is a design parameter and should be given. In order to determine the pressure and temperature after the first high pressure turbine stage, it is necessary to calculate the work done by the fan, the booster and the high pressure compressor. It can be computed by using formulae 2.99 until 2.101. W m˙ c (T T ) (2.99) f an = 2 · p,air t,21 − t,2 W m˙ c (T T ) (2.100) booster = 21 · p,air t,25 − t,21 W m˙ c (T T ) (2.101) hpc = 25 · p,air t,3 − t,25 2.3.E NGINEPERFORMANCE 37

Because the low pressure turbine is assumed to propel both, the fan and the booster, the work calculated from equations 2.99 and 2.100 have to be summed up to get the total work done by the turbine and therefore the correct temperature and pressure. In a similar manner, the high pressure turbine propels the high pressure compressor. To determine the temperature at station 32 use of 2.102 to 2.105 can be made. It is assumed that NGV’s as well as the rotor are actively cooled. This has a major impact on the calculation due to the cold gas stream injected into the flue gas flow. The flue gas flow is therefore first mixed with the cooling air in the NGV. This lowers the temperature before rotor inlet and hence reduces its effectiveness, because the turbine sees a colder temperature than the combustor exit temperature. Due to the mixture also the values of cp will change. The temperature after the turbine is then again a combination from the turbine work and the addition of bleed air. It is however assumed, that the bleed air of the turbine does not have an effect on the turbine performance. Thus, the final temperature at station 32 will be again an average from the turbine cooling temperature, the temperature due to the extraction of work and due to the NGV cooling. The cp value at station 32 is therefore also different from the one between the turbine and the NGV. This can be seen in relations 2.102 to 2.105. m˙ 31cp,g as Tt,31 m˙ cool,NGV cp,air Tcool Tt,31.5 + (2.102) = (m˙ m˙ )c 31 + cool,NGV p,g asmi x1 m˙ 31cp,g as m˙ cool,NGV cp,air cp,g asmi x1 + (2.103) = m˙ m˙ 31 + cool,NGV Whpc Tt,32 Tt,31.5 (2.104) = − m˙ c η 31.5 · p,g asmi x1 · mech m˙ 32cp,g asmi x1Tt,32 m˙ cool,Tur bcp,air Tcool Tt,32 + (2.105) = (m˙ m˙ )c 32 + cool,Tur b p,g asmi x2 m˙ 32cp,g asmi x1 m˙ cool,Tur bcp,air cp,g asmi x2 + (2.106) = m˙ m˙ 32 + cool,Tur b The calculation of pressure is performed in a similar manner. Also here the effect of the pressure of the coolant gas has to be taken into account. The pressure of the coolant should be higher than the pressure in the free stream to ensure a working cooling system. The pressure is then first calculated over the NGV again with a mass average, then the effect of energy extraction due to the turbine is represented in equation 2.102. Afterwards the effect of the turbine cooling air is included. The final pressure at stage 32 can then be retrieved from relation 2.109.

pt,31 pt,31.5 (m˙ 31 m˙ )cp,g asmi x1 (2.107) = m˙ c p m˙ c + cool,NGV 31 p,g as + cool cool,NGV p,air ³ κg as ´ · µ ¶ ¸ κg as 1 Tt,32 1 − pt,32 pt,31.5 1 1 (2.108) = · − − Tt,31.5 · ηi so

pt,32m˙ 32cp,g asmi x1 pcool m˙ cool,Tur bcp,air pt,32 + (2.109) = (m˙ m˙ )c 32 + cool,Tur b p,g asmi x2 Because the influence of the second combustion on the value of κ is not known, it will be assumed, that it stays at the value of κg as for the rest of the cycle. Over the second combustion chamber and turbine the calculations go accordingly. The next step is now to determine what kind of flow is given in the nozzle at the exit. In case the gas flow undergoes a full expansion to atmospheric pressure until the nozzle exit, the flow is unchoked. If the pressure at nozzle exit is higher than atmospheric pressure the flow is choked. To determine this condition, equation 2.110 is of high importance, for it gives the critical pressure ratio ²CR .   pt,5  1  ²CR  κg as  (2.110) = p =  ( )  CR ³ ³ 1 ´ ³ κg as 1 ´´ κg as 1 1 − − η κg as 1 − i s,noz · + If the nozzle is choked, the following relation holds.

pt,5 ²CR pamb > 38 2.L ITERATURE SURVEY

If however pt,5 ²CR pamb ≤ the nozzle is unchoked. The difference of these two nozzle behaviors can be related to the behavior of pressure and temperature. The following calculations are different for a choked or and unchoked nozzle. In case of a chocked nozzle the calculations have to be performed as follows:

2.3.4. CHOKED NOZZLE Having a choked nozzle implies more computational work, because the pressure has still to be calculated and generally some calculations are different than for the unchoked condition. For these calculations see section 2.3.5. To determine the thrust of the core of a choked nozzle the critical temperature, pressure and density have to be calculated. This can be seen in equations 2.111 to 2.113 µ 2 ¶ TCR Tt,5 (2.111) = κ 1 g as + µ 1 ¶ pCR pt,5 (2.112) = pt,5/pCR µ ¶ pCR ρcr (2.113) = R T · CR With these values it is possible, to calculate the nozzle area and the velocity of the jet as it can be seen in equations 2.114 and 2.115. 1 V (κ R T ) 2 (2.114) 8 = g as · · CR m˙ ACR Anoz (2.115) = = ρ V cr · 8 To calculate the thrust the only quantity missing is the free stream airﬂow speed V . It can be calculated ∞ using equation 2.116. p V M κair R Tamb (2.116) ∞ = · · · With the help of these formulas the thrust of the core can be determined using relation 2.117.

F m˙ (V8 V ) Anoz (pCR pt,2) (2.117) = · − ∞ + · −

2.3.5. UNCHOKED NOZZLE In case the nozzle is unchoked, the procedure is a little bit simpliﬁed. The critical temperature is computed according formula 2.118  κg as 1  µ ¶ − pamb κg as TCR Tt,5 Tt,5 ηi s,noz 1  (2.118) = − · − pt,5

Next step is here to calculate the jet velocity, which can be done as depicted in formula 2.119. q V 2 c (T T ) (2.119) 8 = · p,g as · t,7 − CR Finally the net thrust is calculated using equation 2.120 and 2.116.

F m˙ (V8 V ) (2.120) = · − ∞

2.3.6. CALCULATIONS OF BYPASS AIR FLOW The procedure to calculate the bypass airﬂow is similar to the core computation. The equations from 2.110 on have to be used again with the according physical values in order to obtain the conditions of the bypass ﬂow. 2.3.E NGINEPERFORMANCE 39

CALCULATIONOFCOMBINEDTHRUSTAND SFC To calculate the thrust and the SFC, the bypass and core have to be summed up. Equation 2.121 gives the total net thrust, while equation 2.122 gives the SFC of the engine.

X£ ¡ ¢¤ F m˙ (V8 V ) Anoz pCR pt,2 (2.121) = · − ∞ + · −

m˙ f SFC (2.122) = Ftot Concluding it can be said, that the application of cooling the engine is detrimental to the engine performance on one side, because of the lower hot stream gas temperature. This effect however is well compensated by the higher allowable TIT. In the given case with the heat exchanger, the mass ﬂow used to cool the engine however can be reduced. Even though this still reduces the temperature in the hot gas stream, it reduces also the amount of bleed air being taken from the high pressure compressor. Therefore the overall thermal efﬁciency of the engine will rise.

MODULE IMPLEMENTATION

This chapter will refer to the theory discussed in chapter 2. The design will follow the way of calculation given there. If adaptations to equations were made they will be stated here. This was partially necessary since some equations were given in the state of a differential equation. These had to be discretized ﬁrst in order to be able to implement them in the code.

The design implementation will be split into the three main working areas. Those are the heat exchanger design, the cooling part based on the work done in [11], and a ﬁnal part on the gas turbine performance. In general, the solution outline for the total problem follows the indication given in ﬂowchart 3.1. It was decided to model LNG as a mixture out of the constituents methane and ethane. Like this, more than 96 % of mole fraction of LNG is covered and the effects of having a mixture are included. The decision was made to model LNG as a mass ratio of 90% methane and 10% ethane. According to this mixture relation the physical properties of LNG were evaluated for gas and liquid. The properties obtained by [23] are calculated by the EOS derived by the Groupe Européen de Recherches Gazières (GERG) in 2008 and represent the physical properties of mixtures very accurate. In order to obtain a quick interpolation between the data points, the interpolation method of [44] was used.

For the cooling module the input data as given in table 3.1 was taken, which founds the basis design requirement for the heat exchanger. The values for pressure and temperature of air stem from data for the AHEAD engine. The same holds for the fuel mass ﬂow rate of LNG. The temperature of LNG was also given and is a suitable storage temperature of LNG. The pressure of LNG was set to 52 bar. The pressure of LNG can be easily adjusted in the real application by a fuel pump. Since the critical pressure estimation from [23] resulted in a critical pressure of 53.22 bar, a value below this was taken, in order to ensure computational stability within the heat transfer calculation. Theoretically every pressure between the critical pressure and the pressure in the combustion chamber could be chosen. Taking the pressure at this high value however allows for a larger design space and exploitation of more geometrical arrangements for the heat exchanger due to a higher allowable pressure drop. To ensure operation of the engine, the pressure of the LNG stream always needs to be higher than the air pressure. This indicates a limit of the air pressure of 52 bars. At take-off the pressure can exceed this value. To ensure further operation a second fuel pump switched between heat exchanger and injection nozzle could be used.

At first, an estimation on the outcome of the pressure from the heat exchanger is made. Given the mass flow of flue gas at the turbine inlet as well as pressure and temperature, the cooling module will output the required mass flow of cooling air to cool one blade to the required maximum allowable temperature. For the design this temperature was set to T 1366.15K . This is the maximum operating temperature of Inconel 625, which max = was taken representatively as turbine blade material [45]. Temperature and pressure at heat exchanger inlet are equivalent to the exit conditions of the HPT. Therefore the engine module can provide this data.

With this knowledge and geometrical data, the heat exchanger module calculates the required length of the heat exchanger to arrive at the requested air temperature. Also an output pressure of the heat exchanger is given, based on the calculated length and the input pressure. This pressure has then to be compared to the estimated pressure. If the error is small enough, the data of required cooling mass ﬂow is given to the engine

41 42 3.M ODULE IMPLEMENTATION

Start

User defined parameters,T_hex_req, P_hex_est,T4, P4, T_max_blade

Cooling Mdot_cool_req module

Turbo Engine machinery performance T3,P3 data module

Heat L_hex, exchanger P_hex module

Engine P_hex=P_hex performance _est? no yes module

Thrust, SFC

End

Figure 3.1: Solution procedure for the design problem 3.1.H EAT EXCHANGER 43 performance module. The thrust and Thrust Speciﬁc Fuel Consumption (TSFC) can then be determined to estimate the beneﬁt of the heat exchanger.

Table 3.1: Fluid parameters for the design

Parameter Air LNG m˙ [kg/s] - 0.96 P [bar] 43.12 52 T [K] 890 120

3.1. HEAT EXCHANGER

The heat exchanger design will represent the majority of the work done for this thesis. Therefore this chapter will deal extensively with the thermal design philosophy. In the beginning of the design the basic data has to be analyzed. This data gives a rough estimation of the operating envelope of the heat exchanger to make upfront a clear decision, what the ideal type of the heat exchanger for this application will be. The following information serves as the base for the decision making process: • Operating temperature • Operating pressure • Fluids used • Operating environment • Necessary performance The operating temperature for the design is specified by the inlet temperatures of both fluids. Since the temperature difference between the fluids is always smaller than the inlet temperature difference, this will be the measure for the operating temperature. The limiting factor for the operating temperature is twofold. On the one hand low temperatures will increase the brittleness of materials due to the immobility of material imperfections [46]. On the other hand high temperatures can only be achieved up to a certain point dependent on the melting limitations of metals and the accompanied decrease in tensile strength. Since the maximum temperature and the lowest temperature are 890K and 120K respectively, a plate-type heat exchanger cannot be used. Also the high operating pressure of around 40 atmospheres for air and even more for LNG makes the use of this type of heat exchanger not possible. Since the pressure drop is limited to 4% across the heat exchanger and turbine blade arrangement, a pure shell-and-tube heat exchanger will be difficult to realize, since the baffles usually introduce a significant loss in pressure. Additionally the fluids pose constraints on the exchanger. The heat exchanger will operate as both, a liquid-gas exchanger and a gas-gas exchanger. Especially for the first part, the difference in heat transfer coefficient between the liquid and the gas is quite significant. This could be solved by unilaterally extending the heat transfer surface area of the exchanger. The operating environment of the heat exchanger is the aero gas turbine, mounted to an aircraft. This implies that the heat exchanger should be as compact as possible. Furthermore, it should be as light as possible, in order not to deteriorate the overall performance of the aircraft. Shell-and-tube heat exchangers however are usually very heavy and mainly used on ground based installations in industry. Even though they withstand high pressures and temperatures, their rather low heat transfer area per unit volume and heavy weight sorts them out in the design process. To reach the necessary performance, a shell-and-tube heat exchanger would be very likely too big and heavy as it could be strapped on a gas turbine. For this reason, an extended type heat exchanger seems to be very well suited for the heat exchanger. The high pressure stipulates the use of tubes as fluid guidance. The high difference in heat transfer coefficient that is to be expected from LNG and air favors the use of LNG as inner fluid, and air as outer fluid. Fins are added on the air side of the tubes to compensate for the lower heat transfer coefficient of air. This layout is also the preferred order due to the difference in mass flow rate and density of the two fluids. Air has a higher specific volume than LNG and therefore consumes more space. Since baffles also impose a too high pressure drop, 44 3.M ODULE IMPLEMENTATION a cross breed between a double-pipe exchanger and a finned shell-and-tube heat exchanger seems to make sense. This form of a heat exchanger will therefore be aimed at. Concluding, the following heat exchanger will be designed: • Pure counterflow exchanger • One shell pass, one tube pass • Multiple tubes parallel within one shell • Fins will be added to the outside of the tubes • LNG will be the inner fluid, air the outer fluid

3.1.1. GENERALDESIGNCONSIDERATIONS When choosing which methods to use for the design, the geometry of the exchanger as well as its unique design play a role. As discussed in 2.1.3, the flow phenomena to be necessarily included in the calculation procedure vary depending on the effectiveness aimed for. Furthermore, since the heat exchanger suggestion will be a preliminary design, emphasis is also laid on the quick adaptability and flexibility of the program. There- fore the necessary flow phenomena to be included involve only the change in fluid properties as secondary effect. The flow distortion will be neglected because the effect becomes non negligible for heat exchangers above an effectiveness of 80%. Given the required fuel and air mass flows and the specific heat values of the fluids used in the cryogenic heat exchanger, an effectiveness estimation based on comparable applications can be done. The expected effectiveness of the heat exchanger will thus be less than 70%. This estimation is also dependent on the type of flow arrangement and NTU. While a larger NTU value quickly can increase the effectiveness for counter-flow type heat exchangers, multi-pass and parallel-flow exchangers will hardly reach the value of 70% effectiveness. For the design standard theory like the ²-NTU or the LMTD-NTU is not suited because of the change in fluid properties. Also the phase change introduces a problem in terms of modeling. Therefore an alternative must be found. Stream evolution codes as mentioned in [5] could very well model the change in fluid properties. Due to their proprietary nature however they are out of scope of this work. To still be able to introduce the necessary phenomena in the design a distributed parameter model will be chosen. In this model, the heat exchanger will be divided into many segments, thus offering the possibility to update the fluid properties every step. Furthermore it facilitates the modeling of the phase change process. Ultimately it will be possible to write the entire heat exchanger in one code with fluid condition based calculation. Every section the actual fluid properties will be compared to the limiting conditions with respect to the initiation of boiling. When this point is reached the two phase part will continue with the calculation until the calculated vapor quality is 1. Then it is assumed, that pure gas flow exists and the single phase program is triggered again. This method will therefore produce a length required for a given cross section and fluid properties. The implementation of the code will be discussed in the following. The entire code of the program can be found in the appendix C. The model will be based upon an energy balance. The following assumptions are made in order to come up with a design: • No heat leakage to the surrounding • No flow distortion • No longitudinal heat conduction within the wall • Steady state conditions • No heat sources or heat sinks • Constant sectional fluid properties (no gradients normal to the flow direction) • The flow condition is given by the bulk velocity of the stream • Constant mass flow • Only liquid LNG is absorbing heat during evaporation 3.1.H EAT EXCHANGER 45

• Constant temperature of LNG during evaporation Due to the constraints imposed on the heat exchanger the most suitable approach will be the design according the ﬁrst law of thermodynamics, as explained in section 2.1.2. When discretized, the formulae 2.18 and 2.19 are changed to what is depicted in equations 3.1 and 3.2.

UA ∆T1 (T1 T2) (3.1) = (mc˙ p )1 −

UA ∆T2 (T1 T2) (3.2) = i(mc˙ p )2 − These equations will directly give the change in temperature in Kelvin along one sectional element. The value of i depends on the flow configuration. These two relations are the central equations with which the change in temperature will be modeled along the heat exchanger for two phase as well as for single phase flow. In order to determine the factor UA multiple possibilities and relations are given, all based on different assumptions and for a different degree of accuracy. A relation incorporating the heat transfer coefficient of the inner and outer side as well as the heat transfer properties of the tubes is suited best to represent the physics of heat transfer. The calculation of the overall heat transfer is performed according to relation 3.3 for a heat exchanger without fins, as presented in reference[47]. Adding fins to the tubes will naturally alter this relation to what is depicted in 3.4. In these two relations hi and ho give the heat transfer coefficient on the inner and outer side respectively, kt the thermal conductivity of the tube material and l the length of an individual fin. The amount of fins used is taken into account by the factor No f ins . The influence of geometry on the overall heat transfer coefficient is accounted for by the inner tube diameter ri , the outer tube diameter ro and the length of the section ∆x. · µ ¶ ¸ 1 1 1 ro 1 − UA ln (3.3) = 2πri ∆xhi + 2πkt ∆x ri + 2πro∆xho · µ ¶ ¸ 1 1 1 ro 1 − UA ln (3.4) = 2πr ∆xh 2l∆xh No + 2πk ∆x r + 2πr ∆xh i i + f in f ins t i o o While the calculation of the inner heat transfer coefficient is straight forward, the computation of the outer heat transfer coefficient can become difficult in a multi-tube arrangement. Since no correlations could be obtained to determine the outer heat transfer coefficient over a multi-tube arrangement in longitudinal flow configuration, it had to be modeled. As basis for this model the double pipe heat exchanger was selected, due to the similarity of geometry. The outer shell of this artificial double pipe exchanger was taken to be the diameter of the biggest circle allowable in the cross sectional area of the heat exchanger under consideration of the number of tubes used, circles required and diameters of tubes and shell. In figure 3.2 a view on the artificial shells is given. To complete the model, the mass flow through the artificial shell section was set based on the area of the artificial shell. The value for the Reynolds number was kept the same. By doing so, the local heat transfer coefficient for one tube in an artificial shell was determined. In combination with the total heat transfer surface area of all the tubes within the section, the model provides the heat transfer for the multi-tube layout. A scheme of the calculations done within the heat exchanger section can be found in figure 3.3. In the following the individual parts will be given on how they were implemented in the code.

3.1.2. LIQUID PHASE FLOW The first part of the heat exchanger design concerns with the liquid single phase flow. The theory of singlephase design was already discussed extensively in 2.1.6. The equations shown can be directly applied with the method chosen. Like this, every section will undergo a complete calculation cycle. The sequence of events within the liquid phase part can be retrieved from flow chart 3.4 At first inputs are given to the program. Those include geometrical data as specified by the user, as well as initial fluid state data. Next the pressure drop due to the entrance effect will be determined. This will be done just once. The program then determines all the necessary fluid parameters from the fluid temperature and pressure. The data calculated includes the specific heat at constant pressure, thermal conductivity, density and viscosity. Then flow speed as well as Reynolds and Prandtl numbers for the fluids are determined. In dependence of the geometry the overall heat transfer will be calculated. This calculation involves computing the artificial shell diameter of the heat exchanger, the 46 3.M ODULE IMPLEMENTATION

0.1

0.05

−0.05 Vertical Length [m]

−0.1

−0.1 −0.05 0 0.05 0.1 Horizontal Length [m]

Figure 3.2: Artiﬁcial shells (blue dotted lines)

User inputs (geometry, Liquid Start Two phase Gas Phase Plot results End start values, Phase requirements)

Database physical properties

Figure 3.3: Calculation scheme of the heat exchanger module 3.1.H EAT EXCHANGER 47

Start liquid phase Calculation of pressures Calculation of UA

Input data of geometry Determine Determine and initial hydraulic artificial shell conditions diameter of diameter shell

Calculate pressure Determine drop due to Calculate inner heat entrance effects friction and transfer momentum coefficient Calculate sectional pressure drop on tube side fluid properties Determine outer heat Calculate transfer Calculation of UA friction and coefficient momentum pressure drop Determine fin Compute new on shell side heat transfer temperatures coefficient and wall Calculate ΔP temperature Calculation of pressures Calculate UA

Implement pressure drop due to a bend (at discrete locations)

Save data

LNG saturation no reached?

yes

End liquid phase

Figure 3.4: Liquid phase part of the heat exchanger module 48 3.M ODULE IMPLEMENTATION calculation of the inner and outer heat transfer coefficients as well as the heat transfer coefficient of fins and the wall temperature. With this data known, the change in temperature in the section can be determined. Next the pressure drop along the section has to be determined. The computation of the hydraulic diameter of the section allows the calculation of the momentum as well as the frictional pressure drop. The sum of the two gives the total sectional pressure drop. This calculation has to be performed for the tube side as well as for the air side. While for the tube side calculation the computation of one tube is sufficient, the shell side must take the increase in wetted surface by the fins and the multi-tube arrangement into account. If a bend is present at the given location, a single calculation to determine the pressure drop on tube and shell side is executed. After this the data is saved and a check is performed whether the saturation conditions for LNG are already reached. If this is not the case, the program calculates a next step. In case the saturation conditions are reached, the iteration will stop and the two phase module is started.

3.1.3. TWO PHASE FLOW To be able to calculate the two phase flow, an adaptation had to be executed. This holds for the frictional pressure drop calculations. Since these were given in differential form, they had to be discretized. For a sectional length of dx 0 equation 2.74 can be rewritten into what is depicted in relation 3.5. → ½· 2 µ 2 2 ¶ ¸ µ 2 ¶ ¾ m˙ m˙ m˙ 1/3 m˙ 3 ∆p f ,tp ζl 2 ζv ζl x˙ (1 x˙) ζv x˙ dx (3.5) = 2ρl d + 2ρv d − 2ρl d − + 2ρv d · Beside this change, the formulae given in section 2.1.7 were used without further alteration. A flowchart of the sequential contents of the two phase flow part is given in figure 3.5. When the two phase part of the heat exchanger module is run, first initial parameters are set. Those are e.g. the reduced pressure of LNG or an initially assumed heat flux. Also the output data of the liquid phase part of the module is used. Like for the single phase part, first the sectional fluid parameters will be determined, however Reynolds number will only be calculated for the shell side. Since the correlations used for determination of the two phase flow make use of the single phase liquid and vapor Reynolds number, there is no average Reynolds number calculation needed. The program first computes the heat of vaporization for LNG, followed by a calculation on the mass that is vaporized within the section. Then, the vapor quality and void fraction are determined based on this data. Next the frictional pressure drop and the pressure drop due to acceleration on the LNG side, as well as the pressure drop on the air side will be determined. In the following calculations of the overall heat transfer coefficient, the heat flux on the inside of the tube will be determined. This enables calculations for the inner heat transfer coefficient. Moreover the outer heat transfer coefficient, fin heat transfer coefficient and wall temperature will be determined. As next step based on the overall heat transfer coefficient, the temperature change of the two fluids within the section can be determined. The heat inflow for the next section based on the heat flux is recorded and data is saved. Finally the code checks whether the vaporization is already complete. If this is not the case a next section is entered, else the part will exit and the gas phase part will be loaded.

3.1.4. GAS PHASE FLOW The calculation of the gas phase part is essentially the same as the first one. However in contrast to the pressure drop due to entrance effect, a term for modeling the exit effect on pressure is included. Furthermore the requirement for termination of the module is now whether the required air temperature is reached. After the code is finished, the results will be plotted. A flow chart of the gas phase part can be found in 3.6.

3.1.5. LIMITATIONS Due to the constraints mentioned in the assumptions, the tool will come with several limitations. Depending on the mass flow rates chosen by the designer it can happen, that within one discretized step the LNG will completely evaporate. If the energy entering the section is higher than needed to vaporize the LNG, the program will display an error message. In real life application this would imply a rise in temperature directly after vaporization. The two phase flow program however cannot model this. A further point that can arise when using the heat exchanger module are errors in terms of Input/Output (I/O). When the heat exchanger is designed with unrealistic data (e.g. extremely low or high mass flows), 3.1.H EAT EXCHANGER 49

Start two phase Determine state of Calculation of UA the flow Initialization parameters, data from Determine first single phase Determine heat artificial shell of vaporization diameter Calculate sectional fluid properties Calculate liquid Determine heat and gaseous flux Determine state of flow masses

Calculate inner Calculate accelerational pressure Calculate vapor heat transfer drop quality coefficient

Calculate frictional pressure drop Calculate void Calculate outer fraction heat transfer coefficient Calculate pressure drop of air

Calculate fin Calculation of UA heat transfer coefficient and wall temperature Compute new Temperatures

Calculate UA Determine new heat inflow

Save data

Vapor quality = no 1?

yes End two phase

Figure 3.5: Two phase part of the heat exchanger module 50 3.M ODULE IMPLEMENTATION

Start gas phase Calculation of pressures Calculation of UA Input data of geometry Determine Determine and initial hydraulic artificial shell conditions diameter of diameter shell

Calculate sectional Determine fluid properties Calculate inner heat friction and transfer momentum coefficient pressure drop Calculation of UA on tube side Determine outer heat Compute new Calculate transfer temperatures friction and coefficient momentum pressure drop Determine fin Calculation of on shell side heat transfer pressures coefficient and wall Calculate ΔP Implement pressure temperature drop due to a bend (at discrete Calculate UA locations)

Req. air no temperature reached?

yes

Calculate pressure drop due to exit effects

Save data

End gas phase

Figure 3.6: Gas phase part of the heat exchanger module 3.2.T URBINEBLADECOOLINGMODULE 51 errors may occur. In such a case a revision of the inputs is necessary.

Depending on the choice of the number of tubes and number of tube rings within the shell, rounding errors can lead to a display error in the layout. It thus can happen that one tube too less is shown than the designer intended. This should be always checked.

3.2. TURBINEBLADECOOLINGMODULE

As explained in previous chapters the cooling module will be basically overtaken by the code provided by Tiemstra [11]. It however comes with several adaptations in order to be implemented into the code of the heat exchanger. First of all the code is triggered normally over a GUI. Since this is not goal leading in this case due to the many parameters available, a design for a blade was ﬁxed. This enables a comparison of the heat exchanger performance of different conﬁgurations. The blade geometry can be found in tables 3.2 to 3.4.

Table 3.2: Leading edge cooling parameters Table 3.3: Centerline cooling parameters

Description Parameter Description Parameter Cooling method Jet impingement Cooling method Convection Arrangement Staggered No. of channels 4 J.I. Area A 2500 Enhancement method Ribs J.I. Area R 20 Enhancement shape Continuous Parameter D 0.5 V-Shape No Parameter X-distr. 5 Type 5 Parameter Y-distr. 5 Parameter e/D 0.0625 Parameter Z-distr. 5 Parameter P/e 10 Parameter angle 60 Parameter AR (Channels 1-3) 0.5 Parameter AR (Channel 4) 1 Parameter H(mm) 3

Table 3.4: Trailing edge cooling parameters

Description Parameter Cooling method Convection No. of channels 0 Enhancement method Pin ﬁns Enhancement arrangement staggered Parameter D 1 Parameter S/D 2.5 Parameter X/D 4 Parameter w 15 Parameter L 60 Parameter H/D 1

These parameters were isolated from the code and saved. During the calculation of the required cooling mass ﬂow they are loaded into the program before the calculation is executed. Like this, it is not necessary to always launch the GUI. For both, vane and rotor, the same cooling layout is assumed. The maximum allowable temperature for blade and vane was set to 1366.15K . This is the maximum service temperature of the Inconel 635 alloy [45]. 52 3.M ODULE IMPLEMENTATION

Figure 3.7: Engine simulation layout

3.3. ENGINE PERFORMANCE

Because the heat exchanger has its design point at take-off, the engine performance also has to be determined at this point. Since the design point calculations given in section 2.3 do not hold for off design conditions, another solution has to be found in order to get an accurate prediction of the engine performance. During off-design operation the performance of turbomachinery deviates significantly from the performance predicted by design point calculations. This change in performance is non-linear and can only be determined when having compressor and turbine maps. Those however are almost always confidential information of the turbomachinery manufacturer. Therefore generic turbine and compressor maps were used for this thesis. A layout of the engine in GSP can be found in figure 3.7 For the GE-90-94B engine the design condition is top of climb. This is the highest altitude with a full thrust condition applied. The operating conditions for the engine are depicted in table 3.5 and were taken from reference [16].

Table 3.5: Engine operating conditions at design point GE-90-94 [16]

Description Parameter Altitude [m] 10668 Ambient pressure [bar] 0.239 Ambient temperature [K] 218.82 Mach number [-] 0.85

The design point parameters of the engine are depicted in table 3.6. These values were used as an input to determine the design condition of the engine. The heat exchanger however has its design point at sea level with maximum mass flows. This means static sea level at full thrust applied. By doing numerical iterations GSP calculates the engine performance parameters at sea level static conditions. The operating conditions for the engine at this off design condition are depicted in table 3.7. In order to determine the difference in performance between the engine with or without the heat exchanger mounted a bleed sweep was executed. The bleed mass flow for cooling was therefore varied. The according levels of thrust, SFC and fuel flow were then recorded. From the turbine blade cooling module the cooling mass flow requirements with heat exchanger and without heat exchanger mounted to the engine is known. A direct comparison can hence be made between the performance levels of the engine with and without heat exchanger. These results will be shown and discussed in chapter 5. 3.3.E NGINE PERFORMANCE 53

Table 3.6: Engine operating conditions at off design point GE-90-94B

Description Parameter Intake efficiency [-] 0.98 Fan polytropic efficiency [-] 0.93 Compressor polytropic efficiency [-] 0.91 Turbine polytropic efficiency [-] 0.93 Isentropic nozzle efficiency [-] 0.95 Mechanical efficiency [-] 0.99 Combustion pressure loss ratio [-] 0.05 Ram pressure ratio [-] 1.590 Fan pressure ratio [-] 1.650 Low Pressure Compressor (LPC) pressure ratio [-] 1.650 HPC pressure ratio [-] 21.5 Overall pressure ratio [-] 40.44 Bypass ratio [-] 8.1 TIT [K] 1380 Air mass flow [kg/s] 576 Fuel mass flow [kg/s] 1.079

Table 3.7: Engine operating conditions at static sea level GE-90-94B

Description Parameter Altitude [m] 0 Ambient pressure [bar] 1.01325 Ambient temperature [K] 288.15 Mach number [-] 0.00

VERIFICATION &VALIDATION

In order to properly address the topics of Verification and Validation, the terms have first to be defined in an adequate manner. According to the Project Management Body of Knowledge (PMBOK) guide, the terms are defined as follows [48]:

• "Validation. The assurance that a product, service, or system meets the needs of the customer and other identiﬁed stakeholders. It often involves acceptance and suitability with external customers. Contrast with veriﬁcation."

• "Veriﬁcation. The evaluation of whether or not a product, service, or system complies with a regula- tion, requirement, speciﬁcation, or imposed condition. It is often an internal process. Contrast with validation."

According to this definition the verification and validation process will be executed. For the development of the cooling system this means that the validation will ensure, that the result actually meets the design needs, meaning that the heat exchanger will lower the fuel consumption and/or increase the thrust of the engine. Verification will require to assess the modules and verify, that they indeed reflect reality to the degree as specified in the assumptions. From this definition validation and verification have to be performed for different parts of the thesis. While the engine module has to be validated and verified, the heat exchanger module has to be verified only. The cooling module was already verified and validated in previous work done [11].

4.1. HEAT EXCHANGER

Since the heat exchanger is only indirectly an engine performance relevant part, it makes no sense to perform a validation at this point. The only way to validate the design would be to build it and measure the outcome of the heat exchanger. This is however out of scope of this thesis. Because of this it is important to verify that the heat exchanger reﬂects the reality to the degree as speciﬁed in the assumptions.

4.1.1. VERIFICATION In order to assess the heat exchanger module multiple approaches can be used. The one followed in here is the assessment of all the individual subparts of the code. It is then assumed that the combined part is valid as well. Due to a lack of data however not every sub part can be veriﬁed. It also is not possible to verify the complete program in all detail since no reference values could be obtained. For this a comparable tool would be required.

The following parts of the main program should be veriﬁed:

1. Calculations of the overall heat transfer coefﬁcient

2. Calculations of the pressure drop

55 56 4.V ERIFICATION &VALIDATION

3. Two-phase heat transfer coefﬁcient

4. Two-phase frictional pressure drop

5. Two-phase pressure drop due to acceleration

6. Two-phase void fraction

OVERALL HEAT TRANSFER COEFFICIENT

To begin with, the first point will address the verification of the complete module by verifying the overall heat transfer coefficient. Since the heat exchanger is theoretically based on a double-pipe exchanger, this type will be chosen for verification. Reference [17] will provide the necessary data and the comparison calculation. In table 4.1 the input data can be found. Table 4.2 will provide the results.

Table 4.1: Input parameters overall heat transfer coefﬁcient veriﬁcation, example 23-1, [17]

Quantity Fluid 1 Fluid 2 Name [-] Water Oil Diameter [m] 0.02 0.03 Temperature [K] 318.15 353.15 Mass ﬂow [kg/s] 0.5 0.8 Density [kg/m3] 990 852 Thermal conductivity [W/m.K] 0.637 0.138 Prandtl number [-] 3.91 490 Kinematic viscosity [m2/s] 0.602e-6 37.5e-6

Table 4.2: Result comparison overall heat transfer coefﬁcient veriﬁcation, [18]

Quantity Book solution HEx Error [%]

Nuwater [-] 240.60 270.35 12.36 Nuoil [-] 5.25 3.66 69.71 hwater [W/m2.K] 7663.00 8610.70 12.36 hoil [W/m2.K] 75.20 50.51 32.83 U [W/m2.K] 74.50 50.21 32.60

The partly large deviation of the calculation stems from the difference in Nusselt number. In the given case the flow is laminar. For the constant heat flux assumption in the HEx module this results in a Nusselt number of 3.66 for laminar flow [28]. The difference in the Nusselt number on the water side origins in the different correlations used. The reference makes use of the well known Dittus-Boelter correlation as described in reference [49]. For the heat exchanger program however the Gnielinski correlation was adopted as given in [50]. Reason for this is the bigger range of accuracy of the Gnielinski correlation. While the Dittus and Boelter correlation is valid for Reynolds numbers above 10000 only, the Gnielinski correlation can predict Nusselt numbers accurately for Reynolds numbers 3000. ≥

CALCULATIONSOFTHEPRESSUREDROP

As mentioned in the results section, the pressure drop results are flawed due to a program bug that was detected too late. Therefore the pressure drop in general will be lower than the estimation given. After fixing this error it however is possible to verify the calculations based on a comparison with verified data. The frictional pressure drop at the exit station of the first single phase flow should be close to those of the two phase frictional pressure drop at the entrance of the two phase flow. Indeed, by comparing the equations 2.71 and 2.43 with each other it becomes clear, that the build up of the equation is basically similar, except for the case, 4.1.H EAT EXCHANGER 57 that Mueller-Steinhagen and heck make use of the pipe diameter as reference length, while Shah uses the hydraulic radius. This is however compensated by Mueller-Steinhagen and Heck using an own representation of the Darcy friction factor. This is the reason for the small difference of the two methods used. In figure 4.1 the different friction factors that were used can be seen. While the fanning friction factor for the single phase pressure drop was determined by dividing the friction factor of the Moody diagram by a factor of four, the Darcy friction factor was approximated by Mueller-Steinhagen and Heck by a power function. A difference can be seen especially in the laminar flow regime, where there is a discontinuity in the moody diagram. De- spite this error the results are the same. Since the same pressure drop methodology is applied to the air gas stream the pressure drop can be regarded as verified.

0.07 Moody diagram 0.06 Mueller−Steinhagen and Heck

0.05

0.04

0.03

Darcy friction factor 0.02

0.01 0 2 4 6 8 10 4 Reynolds number x 10

Figure 4.1: Comparison between friction factor used by Moody and Mueller-Steinhagen and Heck

TWO PHASE HEAT TRANSFER COEFFICIENT An important part of the work is the calculation of the two phase heat transfer coefficient. The correlation used by Liu and Winterton as derived in [35] however does not contain explicit data for comparison. The authors state, that the correlation was derived from experimental data points given in reference [51]. The data found in the respective papers however is very scattered and partially incomplete. Additionally the individual error of the correlation with respect to the experimental data could not be found. Rather, an overall deviation over the entire range of data was given. A further point of interest is the fact that the fluids used are mostly refrigerants. Due to the age of these publications it was however not possible to get the papers. Since no data could be retrieved to reflect the output of the correlation, several other correlations were used and implemented in the heat exchanger module to be able to make a quantitative comparison. All of them were used with the same input parameters which can be found in table 4.3. The heat flux varies along the evaporation and was therefore kept constant for all of the correlations to eliminate a source of error. The comparison of the two-phase heat transfer coefficients of four different correlations can be found in figure 4.2. The correlations all make different assumptions and have different ranges of validity. In the researched cases however this range is not explicitly stated. This makes it difficult to assess the accuracy of the individual method. In appendix A the different correlations are elaborated on more in depth.

Table 4.3: Inputs for evaluating the different heat transfer correlations

Parameter Quantity

TLNG,inlet [K] 200.20 PLNG,inlet [bar] 51.99 rinner [m] 0.013 m˙ [kg/s] 0.0160 G [kg/m2.s] 30.14 58 4.V ERIFICATION &VALIDATION

Two phase heat transfer coefficients comparison 10 .k] 10 2 Liu−Winterton Kandlikar

8 Gungor−Winterton 10 Bertsch

6 10

4 10

2 10

Two phase heat transfer coefficient [W/m 0 0.2 0.4 0.6 0.8 1 Vapor Quality

Figure 4.2: Two-phase heat transfer coefﬁcients based on different correlations

Comparing the different correlations, it is interesting how large the deviations between the individual correlations are. The offset of sometimes more than 100% with respect to other correlations however is not unexpected. Piasecka reports in her review of the correlations available that the mean average error in heat transfer coefficient can go up to almost 100%, depending on the flow orientation [52]. Additionally the range of validity is an important factor. Besides the quantifiable parameters, the different correlations are usually developed based on a data set including only one or two fluids. These fluids are then in general tested within one channel of different geometry and orientation. Parameters like surface roughness, shape and orientations are usually not included. In most of the papers a general correlation is suggested. Hence it is ex- pectable, that these correlations work fine when used for the fluids from which they were created. For other fluids however often large discrepancies in the predicted heat transfer coefficient can be detected. Therefore this seems to be a problem of research and a lack of a truly general correlation. Fortunately the difference in predicted heat transfer coefficients has negligible influence on the overall outcome of the HEx module, since the heat transfer coefficient of the two phase flow plays a minor role in the calculation of the overall heat transfer coefficient.

TWOPHASEFRICTIONALPRESSUREDROP

In order to verify the two phase frictional pressure drop, a direct comparison between the results of Mueller- Steinhagen and Heck can be made. In their paper on the development of the correlation for determining the frictional pressure drop [19], examples are given for a two-phase mixture of air and water. The exact parameters used can be found in table 4.4. Additionally, the mixture used is validated by experiments given in the same paper. The calculations are given for several pipe diameters. In ﬁgure 4.3 and 4.4 a comparison between the results of Mueller-Steinhagen Heck and the heat exchanger module are given. Figure 4.4 is taken from reference [19], ﬁgure 2. The calculations were performed for a constant mass velocity with different pipe diameters. The upper limit of the shaded are in graph 4.3 gives the lower pressure while the lower bound gives the upper pressure. Since it is not known which line corresponds to what pressure it was decided to compare the results based on a range.

It can be seen from these figures, that the heat exchanger module resembles the results given by Mueller- Steinhagen and Heck very well. Even though no quantitative numbers are provided by the author, visual inspection confirms the agreement of the two results. It should be mentioned, that the temperature was not given my Mueller-Steinhagen and Heck and was arbitrarily set to T 288.15K , since this is the International = Organization for Standardization (ISO) reference value. Since this influences density as well as viscosity, this will have an influence on the results. The good agreement between the data of Mueller-Steinhagen and Heck 4.1.H EAT EXCHANGER 59

Table 4.4: Input parameters frictional pressure drop veriﬁcation, [19]

Parameter Quantity Pressure [bar] 1.2-1.6 Mass velocity G[ kg ] 200 m2s Density water ρ[kg/m3] 999.11-999.13 Density air ρ[kg/m3] 1.45-1.94 Viscosity water µ [mPa s] 1.1376-1.1375 Viscosity air µ [mPa s] 0.018005-0.018012

Figure 4.3: Results of the heat exchanger module Figure 4.4: Results by Mueller-Steinhagen and Heck method 60 4.V ERIFICATION &VALIDATION

0.8 α 0.6 CH4 0.4 C2H6

Void Fraction C3H8 0.2 C4H10 C5H12 N2 0 0 0.2 0.4 0.6 0.8 1 Vapor Quality

Figure 4.5: Void fraction over vapor quality with the calculations however allow the frictional pressure drop to be regarded as veriﬁed.

TWOPHASEPRESSUREDROPDUETOACCELERATION Being the main contribution of the two phase pressure drop, the pressure drop due to acceleration needs veriﬁcation. However no data for comparison could be retrieved. Additionally this topic Thus this part could not be veriﬁed and thus offers a potential source of error. Due to the low overall pressure drop of LNG it is however believed, that the possible introduced error and the effect on the design is very low.

VOIDFRACTION The void fraction in the program is determined by the principle of minimum entropy production by Zivi [40]. The results discussed in chapter 5 indicate however a slower rise in void fraction than expected. For this reasons this part has to be verified. The constituents of LNG were plotted as pure substances versus the vapor quality. This can be seen in figure 4.5. As it can be seen a void fraction of above α 0.8 may be reached at = a vapor quality of x˙ 0.08. After this rapid increase the void fraction increases slowly until it reaches unity. = When the void fraction of figure 5.5b) is compared with figure 4.5, it becomes clear, that there is a great difference. While the results of figure 4.5 show a relatively steep rise as it can also be found for several substances in literature, the void fraction of 5.5b) is much more flattened. The reason for this can be found in the fact, that figure 5.5b) represents LNG as a mixture, while figure 4.5 uses pure fluids. The main characteristic of the void fraction comes from the difference of specific volume between gas and liquid. The influence of the individual constituents of LNG onto each other seems to alter this in a nonlinear way. Furthermore the effect of high pressure and thus a relatively low density difference between the gas and liquid has an influence on the void fraction. Furthermore a low mass velocity seems to have a flattening effect. This can be seen in graph 17.8 from reference [53]. In this example refrigerant R-410A was calculated by the method of Rouhani for different mass velocities. Unfortunately no quantitative verification could be performed due to a lack of data for comparison. However the qualitative discussion indicates that the program correctly represents the void fraction.

4.2. ENGINE SIMULATION

Veriﬁcation of the engine ensures that no modeling errors were done within GSP. Further validation will show, whether the updated engine design indeed fulﬁlls the customer’s need. 4.2.E NGINE SIMULATION 61

Table 4.5: Design condition engine veriﬁcation GE-90-94AB, [18]

Quantity Reference GSP Relative error [%]

Tamb [K] 216 216 0.00 T2 [K] 243.65 243.65 0.00 T21 [K] 281.61 281.76 0.05 T13 [K] 281.61 281.76 0.05 T25 [K] 313.3 313.35 0.02 T3 [K] 772.55 754.00 2.40 T4 [K] 1450 1450 0.00 T45 [K] 1053.86 1080.05 2.49 T5 [K] 731.14 759.38 3.86 T8 [K] 627.58 645.8 2.9

Pamb [Pa] 22,632 22,632 0.00 P2 [Pa] 34,498.86 34,498.86 0.00 P21 [Pa] 54,760.97 55,198.18 0.80 P13 [Pa] 54,760.97 55,198.18 0.80 P25 [Pa] 76,665.36 77,277.45 0.80 P3 [Pa] 1,456,641.8 1,468,271.61 0.80 P4 [Pa] 1,388,376.12 1,409,540.75 1.52 P45 [Pa] 337,999.90 352,525.13 4.30 P5 [Pa] 66,139.36 73,132.25 10.57 P8 [Pa] 35,256.76 37,797.93 7.21 Total Thrust [kN] 76.07 77.24 1.54 SFC [g/kN.s] 13.86 15.30 10.39

4.2.1. VERIFICATION In order to verify that the engine simulation tool reflects the actual engine behavior, two reference data sets were used. Since the engine will operate at off-design condition this case needs to be evaluated. In contrast to the heat exchanger, the design condition for the engine is the cruise phase at altitude. As discussed in chapter 3, it is not possible to adequately model off-design conditions of gas turbine engines by analytical calculations. Therefore the tool GSP was employed to determine the off-design engine behavior. In order to produce valid data for the off design condition, the design condition was verified with reference data. Since the two reference data sets show a certain variation in engine input parameters, the engine model was adapted in terms of those parameters, to allow a comparison. A comparison between the output data of reference [18] and the data produced by the engine simulation module can be taken from table 4.5. Since for this reference only design point data was given (however very detailed), no off design condition verification can be performed. The second reference for the engine module stems from [16]. The comparison between the design point data of the engine simulation module and the reference can be retrieved from table 4.6. From this reference only Thrust levels and SFC can be retrieved.

Table 4.6: Design condition engine veriﬁcation GE90, [16]

Quantity Reference GSP Relative error [%] Total Thrust [kN] 69.20 75.76 9.48 SFC [g/kN.s] 15.6 14.7 5.77

The discrepancy in the design point calculations from between the program and [18] are attributed to the fact, that in the hand calculations performed the speciﬁc heat is assumed constant, while in reality it changes with temperature and pressure. The discrepancy to the results given in[16] cannot be explained, since the author only states that a program was used for the calculations, however does not give the program. Moreover the 62 4.V ERIFICATION &VALIDATION assumptions done for the calculations are not stated. Furthermore not all of the parameters given could be implemented in GSP. Still, the outcome of thrust and TSFC are in limits of the known engine performance. From this model an off design condition was derived, which in contrast to the design point calculates engine performance at sea level and no air speed. This simulates a take-off condition, with full thrust applied. The engine performance data for off-design condition can be found in table 4.7.

Table 4.7: Off-design condition engine veriﬁcation GE90, [16]

Quantity Reference GSP Relative error [%] Total Thrust [kN] 375.3 427.34 13.87 SFC [g/kN.s] 7.90 7.90 1.27

The off-design condition show a particularly large deviation in thrust. As for the design condition calculations, it however cannot be retrieved what causes the difference. Different calculation algorithms and pro- grams can result in this error. The outcome of SFC agrees well this reference.

4.2.2. VALIDATION This section will deal with the validation process of the engine module. The goal of implementing the heat exchanger is to save fuel and increase the thermodynamic efﬁciency. Therefore in this section a comparison will be made between the gas turbine with applied heat exchanger and without. It will thus address a macroscopic perspective of the entire system.The comparison is done in a qualitative way, since the question whether or not the heat exchanger meets the needs for which it was designed can be answered with a simple ’yes’ or ’no’.

The performance parameters assessed are thrust and SFC. These two parameters are of most interest for the aircraft operation. As already given in the section on results 5, the net effect of the heat exchanger is a gain in thrust and a reduction of TSFC. When regarding the system as a whole, additional concerns like increase in complexity, reliability, maintenance intensity, weight, environmental impact and cost arise as well. This entire envelope cannot be analyzed, also since the analysis of the heat exchanger shows only the beneﬁts at take-off conditions during steady state operation. In order to cover the whole range more information is needed. It can however be estimated, that cost and weight of the engine will increase. The maintenance effort that is needed for the heat exchanger is estimated low, since pure ﬂuids with no impurities are taken for operation. The system complexity will also increase only to a small fraction.

An interesting aspect is as well to determine the heat exchanger performance and benefit during the cruise phase. Since this however is already out of scope of the work it will be discussed only shortly. With the available data of the AHEAD engine it was found out that the relation between air mass flow required for cooling and the LNG stream changed. Due to a higher burning temperature in the LNG combustion chamber of the AHEAD engine with respect to the previously modeled GE-90-94B engine, the required cooling mass flow will be higher. The reduction in temperature at cruise altitude is contributing positively to the cooling performance and thus reducing the need for cooling air. On the other hand, the LNG mass flow is reduced. This leads to a change in mass flow ratios for the heat exchanger. Because of the increase in air mass flow and the reduction in LNG mass flow the two phase part will be short. This would logically imply that the heat exchanger effectiveness should reduce. However the effect that the specific heat rate ratio changes as well leads to an increase in effectiveness of the heat exchanger.

Using AHEAD data, the overall effectiveness of the heat exchanger during cruise is ²=59 %. In order to be able to compare the cruise data with take-off data, the pressure of LNG was reduced to 52bar. This was necessary since the air pressure at take-off for the AHEAD engine lies above the critical pressure of LNG. With this data a take-off effectiveness of the heat exchanger of ²= 49% was determined. Thus it can be expected, that thrust will increase and SFC will reduce during cruise ﬂight as compared to take-off. Thus the performance can be regarded to be validated from this aspect.

From chapter 3 the weight of the exchanger was determined to be M 158.13kg. Since three units are = used for the heat exchanger the total weight will increase by M 474.39kg. Compared to the weight of = 4.2.E NGINE SIMULATION 63

M 7,550kg of the GE-90-94B [54], which is the baseline engine of the AHEAD engine, this means a weight = increase by 6.28%. This estimation however does not include piping and other adaptations that have to be done on the engine. Overall the weight is however not estimated to increase by more than 10 %. The effect on overall performance is however different depending on aircraft used and cannot be further analyzed. It is however believed, that the addition of the heat exchanger will positively contribute to the aircraft operation.

RESULTS &DISCUSSION

This chapter will discuss the results of this thesis based on the ﬁndings from the three modules. The explicit design suggestion will be presented in chapter 6, which will give a suggestion for the geometry of the exchanger based on the thermal outputs.

5.1. INPUTS

In order to estimate the results a discussion ﬁrst on the input data with which the results were produced is given. Like this it is easier to get an overview on the basis of these results. It is also needed, since the inputs stem from different sources. Thus is can be confusing if it is not stated explicitly which input data was used to obtain the results.

This section will ﬁrst deal with the inputs for the heat exchanger module. Next, cooling module inputs will be presented. The section closes with the input data for the engine performance module.

5.1.1. HEAT EXCHANGER

The heat exchanger was designed for the flow conditions depicted in table 5.1. The geometry was chosen to meet the specifications of the input requirement. Since LNG as well as air were regarded as clean fluids, no fouling was dealt with when designing the geometry.

Table 5.1: Fluid input requirements

Description Parameter Input temperature air [K] 890 Output temperature air [K] 600 Input temperature LNG [K] 120 Input pressure air [bar] 43.12392 Input pressure LNG [K] 52 Mass ﬂow LNG [kg/s] 0.96

Additionally the mass ﬂow of air was needed, but unknown. It had to be taken from the cooling results 5.3. From the calculations done in the cooling section it was determined to be m˙ 4.07kg/s. Based on this cool,air = information the heat exchanger geometry was designed.

65 66 5.R ESULTS &DISCUSSION

5.1.2. COOLING

For the cooling module the inputs are derived from the work done by Tiemstra as described in [11]. As input to the cooling module therefore the inputs of the cooling module can be seen. Additionally the data regarding the layout of the HPT stage is needed. This data can be found in table 5.2.

Table 5.2: Fluid input requirements

Description Parameter TIT [K] 1450 Pressure after combustion [bar] 41.399 Cooling air temperature [K] 600 Maximum service temperature blade material [K] 1366.15 Number of stages 1

Missing from this set of data is the pressure of the cooling air. It had to be extracted from the data of the heat exchanger module. After iteration it was found to be p 43.085bar . The cooling arrangement for the ﬁrst cool = and second stage of the HPT was assumed to be the same. Additionally the cooling arrangement for the rotor and vane was assumed to be the same. This includes the number of blades as well as their internal cooling layout. Taken from reference [11] was the number of turbine and stator blades to determine the amount of cooling mass ﬂow needed over the entire HPT. The respective values are listed in table 5.3.

Table 5.3: Turbine parameters

Description Parameter Number of turbine blades per stage 144 Number of NGVs per stage 80 Number of stages 2 Total amount of blades 448

5.1.3. ENGINE PERFORMANCE

The inputs for the engine performance calculations were already given in table 3.7. Additionally to those it is necessary to deﬁne the share of the total cooling mass ﬂow that is assigned to turbine rotor and stator. The cooling air distribution was made according to the amount of blades as mentioned in table 5.3. Thus 64 % of the cooling air is used to cool the rotor and 36 % is required for cooling the NGV.

5.2. RESULTS FROM HEAT EXCHANGER MODULE

The final geometrical cross section can be found in figure 5.1. Given this cross sectional area the total thermal length required by the heat exchanger to fulfill the requirements given in table 5.1 is L 2.46m. This implies = using three parallel units strapped onto the gas turbine. With this layout the effectiveness of the exchanger is ² 39%. The geometrical data specifications of the heat exchanger can be found in table 5.4. = 5.2.R ESULTS FROM HEAT EXCHANGER MODULE 67

0.1

0.05

−0.05 Vertical Length [m] −0.1

−0.1 −0.05 0 0.05 0.1 Horizontal Length [m]

Figure 5.1: Cross sectional view of the heat exchanger

Table 5.4: Heat exchanger geometrical speciﬁcations

Description Parameter Length(straight pipe) [m] 2.46 Outer shell diameter [m] 0.26 Inner tube diameter [m] 0.026 Wall thickness tube [m] 0.001 Number of units [-] 3 Fin length [m] 0.007 Fin width [m] 0.0009 Number of ﬁns per tube [-] 16 Number of tubes [-] 20 Number of tube rings [-] 2

5.2.1. HEAT TRANSFER RESULTS

The temperature distribution and heat transfer along the heat exchanger is given in figure 5.2. As it can be seen, the layout complies well with the requirements. It can be seen that the temperature of LNG first starts rising from the outgoing point of T 120K on. Then at location L 0.84m the evaporation starts. In this LNG = = case it can be seen that the temperature of LNG is constant throughout the evaporation process. The phase change of LNG is then completed at station L 1.15m. The temperature starts rising again until the outlet = temperature of T 378.8K . For the air side the flow begins on the right side of the graph, meaning at LNG = location x 2.46m. There is gas to gas heat transfer and the temperature of the air keeps dropping until the = point of L 1.15m, from which on the temperature drops even faster. This is due to the increased heat transfer = during the phase change and hence due to the evaporation of LNG. After this phase a gas/liquid exchange phase completes the process and the air temperature approaches the required air temperature. The red line in the graph indicates the wall temperature. The wall temperature is a function of the heat transfer surface area, which stays constant and the heat transfer coefficients of the two fluids. The wall temperature shows a small collapse in temperature shortly before the two phase part starts (x 0.65). This can be attributed to = a change in the LNG Prandtl number. Within the Prandtl number definition, the specific heat at constant pressure is responsible for this behavior. During the phase change, the temperature of the wall rises slowly in positive length direction, but is much lower than before. This effect can be attributed to the higher ability 68 5.R ESULTS &DISCUSSION

a) Temperature distribution b) Heat exchanger performance 900 0.4 T Air T 0.35 LNG T 700 Wall 0.3

0.25

500 0.2

0.15 Temperature [K] 300 0.1

0.05 Heat transfer performance U*A [W/K]

100 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Length [m] Length [m]

Figure 5.2: Temperature distribution and heat transfer along the heat exchanger of LNG to absorb heat. After the phase change is completed the wall temperature recovers again and shows shortly after again the behavior from the beginning.

From this it can be concluded that the two phase part should last for as long as possible in order to get the highest effectiveness. By doing so, the temperature change of air is highest and the temperature difference required can be obtained the fastest. For a real life application however other concerns can come up. First of all for the AHEAD engine it is required that the LNG is completely transformed when exiting the exchanger. When the engine is mounted to an aircraft, multiple inlet conditions apply to it. Thus also the input conditions for the heat exchanger will change. This holds also for the LNG mass flow. Different throttle settings will produce different mass flows for the exchanger. Hence it is of importance to ensure that for all throttle settings the heat exchanger produces pure LNG vapor and no liquid LNG is left. From this perspective it might be not best to have the phase change throughout the entire heat exchanger since a small change in LNG mass flow quickly could violate the requirement of having the LNG fully vaporized. Another approach would therefore be to create a very versatile design, which is very robust against change in flow conditions. In such a case the geometry of the heat exchanger could be designed such, that the design mass flows can be found in the center of the operational envelope.

The operational envelope and the effectivenesses will be further discussed further in section 5.5. In the chosen design, the phase change is kept in the middle of the heat exchanger, allowing for a variation in mass flows while still exploiting the benefit of the high heat transfer capabilities during the phase change. In figure 5.3 the heat transfer coefficient of the shell and tubes side can be seen. The heat transfer coefficient on the LNG side first increases due to the increase in Nusselt number. At the same time the thermal conductivity is falling. However the rise in Nusselt number is dominant. At location x 0.65 The influence of the LNG = Prandtl number can be see. The Prandtl number rises at this point. This has an effect on the Nusselt number, which leads to this rise. Due to the discrete nature of treating the three flow regimes, a discrete jump in the tube heat transfer coefficient can be observed. During the phase change the heat transfer is slightly decreasing. Since different models for determining the heat transfer coefficient show different behavior, an analysis of the physics depending on the output of a single correlation is difficult. It however can be said, that the nucleate boiling phenomena is stronger than the forced convection boiling one. This can be deduced, since the nucleate boiling effect is the main contribution of the two phase heat transfer coefficient at low vapor qualities, while forced convection becomes important with and increase in flow speed and thus at higher vapor qualities.

After the phase change is completed, the values of the inner heat transfer coefﬁcients fall again as expected 5.2.R ESULTS FROM HEAT EXCHANGER MODULE 69

a) Heat transfer distribution tube b) Heat transfer distribution shell 6 10 K)] K)] 2 2 132 5 10

128 4 10

3 124 10

2 Inner heat transfer coefficient [W/(m 10 Outer heat transfer coefficient [W/(m 120 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Length [m] Length [m]

Figure 5.3: Heat transfer coefficients along the heat exchanger and a further decrease in Nusselt number leads to a reduction in heat transfer coefficient. The fact that the Nusselt number falls is based on the interplay of a peaking and then falling Reynolds number as well as a falling Prandtl number. The physical behavior of the thermal conductivity of LNG gas leads to a minimum of the heat transfer coefficient at x 1.9. Besides the higher flow speed and the higher Reynolds number the = lower thermal conductivity leads to a lower exit heat transfer coefficient than the inlet heat transfer coefficient. On the right hand side of figure 5.3 the outer heat transfer coefficient can be found. It represents the heat transfer coefficient of the tubes on the shell side as well as the heat transfer coefficient of the fins. The behavior of the heat transfer coefficient of the air side is mainly dictated by the changes in fluid properties. The Nusselt number along the air flow direction is slowly rising, but the change over the entire heat exchanger is low. The changes to be seen in the graph origin mainly the change in thermal conductivity. With decrease in temperature of air also the thermal conductivity drops. Due to the decrease in slope of temperature during the two phase part also the thermal conductivity and hence the shell heat transfer coefficient drops. After this part is finished, the heat exchange between liquid LNG and air again shows a higher slope. As last part of the heat transfer the fins will be discussed. The fin heat transfer coefficient and the temperature distribution of the fin are shown in figure 5.4a) and b) respectively. Similar effects can be seen in this graph as compared to figure 5.3. Since the fluid is the same, this behavior was expected. In subgraph b) the fin temperature distribution can be found along the heat exchanger. The dependency of the fin temperature on the wall temperature can be easily seen. As an adiabatic fin tip is used, the heat transfer rate through the tip is assumed to be zero. Overall it has to be mentioned that the solution might be valid for small scale fins. When the length of the fins gets larger it can be assumed, that longitudinal conduction effects within the fin are not negligible anymore and can lead to a flattening of the temperature slope along the heat exchanger. This effect is likely to be more pronounced at the fin tip as compared to the root.

5.2.2. PHASE CHANGE RESULTS Next, the phase change will be discussed more elaborately. In figure 5.5 three graphs are depicted, giving the thermodynamic equilibrium quality of the flow, the void fraction of the flow and a comparison of the two. From the quality plot it can be concluded, that the phase change in total occupies an exchanger length of L 0.29m. It can be seen, that the quality shows an almost linear increase. This is expected since the = heat flux is relatively constant. Due to this, the amount of vapor produced in each section is also constant. Due to the increase in void fraction with vapor quality, the assumption that only liquid mass will absorb heat during the gasification process however will lead to an increasing error. This effect is estimated to increase 70 5.R ESULTS &DISCUSSION

Figure 5.4: Heat transfer of the ﬁns

a) Vapor quality distribution b) Void fraction distribution c) Vapor quality over void fraction 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 Void fraction Vapor quality Vapor quality 0.2 0.2 0.2

0 0 0 0 1 2 0 1 2 0 1 2 Length [m] Length [m] Void fraction α

Figure 5.5: Vapor quality and void fraction along the exchanger 5.2.R ESULTS FROM HEAT EXCHANGER MODULE 71

Two phase heat flux and superheat temperature 140 100

120 98

100 96 ] 2

80 94

60 92 Heat flux [kW/m 40 90 Superheat temperature [K] 20 88 Heat flux Superheat temperature 0 86 0 0.2 0.4 0.6 0.8 1 Vapor quality

Figure 5.6: Heat flux and superheat temperature along the two phase flow region the necessary length for evaporation for a real life application. Because of the high amounts of gas present in later stages of the phase change it is to be expected that the vapor quality curve slightly flattens out.

The void fraction as shown in the second subgraph shows also a rapid increase. However it is not as steady as the vapor quality. This was expected since the volume occupied by the gas quickly exceeds the volume occupied by the liquid. Still it was expected, that the void fraction would increase faster. This however can be attributed to the high pressures present and the high density of the gas.

The heat flux as calculated by the method described in reference [35] as well as the superheat temperature are given along the two phase flow in figure 5.6. The superheat or excess temperature is defined as described in equation 2.54. It gives the difference between the saturation temperature of the fluid and the wall temperature. The reason for the increase in superheat temperature along the LNG flow direction can be seen in the increase of air temperature. Since the temperature of LNG remains the same, this implies directly that the wall temperature is rising along the LNG flow direction, which is as expected. In total, the superheat temperature is very high. Unfortunately no reference data for comparison could be obtained. The heat flux on the other hand is decreasing along the LNG flow direction. Initially the heat flux is very high due to nucleate boiling. With increasing vapor quality forced convection comes into play which leads to a stabilization in the heat flux.

5.2.3. PRESSURE DROP RESULTS

Next, the pressure drop within the multiphase section of LNG will be discussed. Figure 5.7a) shows the total two phase pressure drop, while 5.7b) gives the individual contributions of the two phase pressure drop. Those are the pressure drop due to acceleration and the frictional pressure drop. It can be seen, that both pressure drops are increasing almost linearly. The pressure drop due to acceleration however rises much quicker than the frictional pressure drop. This indeed confirms, that the pressure drop due to acceleration cannot be neglected. Initially it can be seen from subgraph b), that the pressure drop due to acceleration rises a bit quicker. In the end on the other side it can be seen that the slope is decreasing slightly. This can be explained by the fact that initially the volume occupied by the fluid is comparably large. Thus the implied pressure drop due to the acceleration is also bigger. Approaching a vapor quality of x˙ 1, there is hardly any fluid left. The = main velocity of the flow is defined by the gas phase. As in this given case the fluid velocity of LNG is very low, the difference in flow speed between vapor and liquid is large. This explains also the high values for the pressure drop due to acceleration as compared to the values obtained by the frictional pressure drop.

Due to the fact that at the entrance of the two phase region vapor quality and void fraction are both zero, 72 5.R ESULTS &DISCUSSION the calculation of the pressure drop due to acceleration shows an instability. This is since the first part of the second term in relation 2.68 results in a "NaN" statement of MatLab. The same holds for the exit conditions, in which vapor quality as well as void fraction are 1. This has an influence on the second part of the first term of relation 2.68:

(" 2 2 # " 2 2 #) 2 vg xe,out v f (1 xe,out ) vg xe,in v f (1 xe,in) ∆pa G − − = α + (1 α ) − α + (1 α ) out − out in − in

By approaching the critical values of both cases, it was found that the second part of the first term approaches v f , while the first part of the second term approaches vg . Therefore this value was manually implemented for this special case. This manual implementation of the starting and end values of the pressure drop due to acceleration also may manifest in the results. Due to the analytical behavior discussed above, this effect however is believed to be small. The frictional pressure drop is almost linearly increasing. This is expected since the effect of friction is linear. The transition from liquid to gas introduces a peak in the pressure drop around x˙ 0.85. Because the fluid enters as a liquid and exits as pure vapor, the frictional pressure drop = at the stations x˙ 0 and x˙ 1 is equal to the frictional pressure drop of pure liquid and vapor respectively. = = An important factor in the design of the heat exchanger is also the pressure drop on the air side. Since the

a) Total two phase pressure drop b) Contributions to two phase pressure drop 5 5 Due to acceleration Due to friction 4 4

3 3

2 2

1 Pressure drop [Pa] 1 Total pressure drop [Pa]

0 0 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 Length [m] Length [m]

Figure 5.7: Pressure drop within the two phase LNG flow combustion chamber has a certain percentage in pressure loss, the parallel way through the heat exchanger and the turbine blades is not allowed to exceed this pressure drop. Otherwise backflow will occur, which would damage the engine. In figure 5.8 the total pressure drop along the heat exchanger can be seen for the LNG as well as for the air side. The pressure drop overall for both, LNG and air is low. This has to do with the general layout of the heat exchanger which was chosen among other reasons to minimize the pressure drop. First the air pressure drop will be discussed. It can be seen that the pressure drop of air is fairly constant. This was expected since besides the change in temperature and the accompanying change in physical properties there is no other influence that could create a high pressure drop. When air is injected into the core of the heat exchanger a pressure rise is occurring due to the increase in flow area. The fluid is decelerated and pressure is increased. The pressure drop then builds up along the heat exchanger. The sharp peak at the end of the air flow path comes from the change in cross sectional flow area when the air streams back into an exit tube. The reason for the difference in size of the pressure drop of inlet and outlet can be found in irreversible effects that follow an expansion or compression of a fluid. Along the pressure drop line three kinks can be found. Those are due to the three bends of the heat exchanger (compare figure 6.1). For the LNG stream the behavior is slightly different since there multiple stages are passed on the way into the heat exchanger core. The entrance and exit show also here a different behavior. First of all the LNG stream comes with two entrance pressure changes and two exit pressure changes. The first entrance pressure change comes from the 5.2.R ESULTS FROM HEAT EXCHANGER MODULE 73

a) Pressure drop distribution

4000 Air pressure drop LNG pressure drop 3000

2000

1000

Pressure drop in [Pa] 0

0 0.5 1 1.5 2 2.5 Length [m]

Figure 5.8: Pressure drop along the heat exchanger change in area when LNG is flowing from the tubing into the heat exchanger head. The second change arises when LNG flows into the core. Thus there is first a pressure rise followed by a pressure drop. The entrance effect is however not very pronounced. This can be attributed to the fact that LNG enters as a liquid and also that the change in cross sectional flow area from the inlet tubing to the tubing in the heat exchanger core is small as compared to the area change the air stream undergoes. After the entrance the pressure drop is slightly increasing due to a decrease in density. At station x 0.84 the phase change starts. The pressure there = drops and stays relatively small until the phase change is finished. At phase change exit the pressure drop rises again and keeps on rising until the exit. The higher pressure drop can be attributed to the fact that the density of the gas is lower, hence the flow speed is increased. At the exit, again a small rise in pressure drop can be seen due to the change in cross sectional flow area. The influence of the bend peaks on the LNG stream is also lower than on air. This is because the relation between the bend curvature and the tube diameter is much bigger. Hence the bend pressure drop is of lower influence.

Finally it has to be said that a programming mistake inﬂuencing the pressure drop outputs was found. Since the friction factor was calculated in a wrong way the overall pressure drop in the single phase regions is too high by roughly a factor of twenty. This means that the pressure drop on the air side as well as in the liquid and gas phase calculations on the LNG side are affected. The error was found out as the frictional pressure drop at the two phase entrance should roughly coincide with the exit pressure drop of the liquid phase part. Accordingly the frictional pressure drop at the end of the two phase part where the vapor quality is almost 1 should be close the gas phase pressure drop at entrance of the second single phase part in the LNG stream. Since the friction factor linearly affects the pressure drop, the error in friction factor is proportional to the error in pressure drop.

5.2.4. FLOW PARAMETER RESULTS

In figure 5.9, the flow parameters along the heat exchanger are given. In subfigure a) the Reynolds numbers along the heat exchanger is given. The Reynolds number of air increases along the air flow direction from Re 17,290 to a final value of Re 22,410. The slope is stronger within the phase change part. This is due to = = the change in density. This effect is slightly dampened by a decrease in flow speed. For LNG multiple graphs are shown. The green line in subfigure a) refers to the initial liquid LNG. In red, the liquid LNG Reynolds number during the phase change is shown. The turquoise color indicates the vapor Reynolds number during the phase change. Finally the purple line displays the Reynolds number of the gaseous LNG. Initially the Reynolds number is rising due to an decrease in viscosity, having a minimum at x 0.82. Due to the constant = temperature during the phase change the properties stay constant in this part. After the gasification process is complete, the Reynolds number keeps increasing, attributed to the same effect.

In subgraph b) the Prandtl number can be seen. The Prandtl number for air is fairly constant over the entire exchanger. It starts at a value of Pr 0.72 and reduces slightly to a value ofPr 0.71. For LNG a more = = nonlinear behavior can be observed. Especially in the liquid region, the ratio between the momentum and 74 5.R ESULTS &DISCUSSION

4 x 10 a) Reynolds number distribution b) Prandtl number distribution 8 6 Air Air LNG LNG 6 SP1 SP1 LNG LNG TP,Liq 4 TP LNG LNG 4 TP,Gas SP2 LNG SP2 2

2 Prandtl number Reynolds number

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Length [m] Length [m] c) Flow speed distribution d) Density distribution 2 500 Air Air LNG LNG

] 400 1.5 3 LNG gas 300 1 200

0.5 Density [kg/m Flow speed [m/s] 100

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Length [m] Length [m]

Figure 5.9: Distribution of Reynolds number a), Prandtl number b), ﬂow speed c), and density d) along the heat exchanger

thermal diffusivity changes rapidly. A minimum can be observed at location x 0.36. At x 0.8 a peak is = = observed. This can be attributed to the following effects. First of all the viscosity is falling, but starts at a maximum. The value of c on the other hand starts at small values and rises until a peak at x 0.81. In the p = two phase part also here the physical properties do not change due to the constant temperature and almost constant pressure. In the gas phase part of the LNG stream the behavior of the Prandtl number is dominated by the speciﬁc heat at constant pressure which quickly decreases. This behavior is slightly dampened by a rising viscosity.

In subgraph c) the flow speed along the heat exchanger can be seen. The air flow is slightly decreasing along the air flow direction due to the increase in density with a decrease in temperature. For LNG on the other hand the flow speed is initially very low due to the high density. Due to the gasification process however flow speed increases. The velocity development is mainly attributed due to the change in density, as depicted in subfigure d). Since mass flow is kept constant and no change in cross sectional area is made, this makes sense. Also at the low speeds there is no need to take compressibility effects into account.

5.3. RESULTS COOLING MODULE

The results obtained from the cooling module can be found in table 5.5. It can be seen that the results for an individual blade or vane is the same as expected since the same layout was chosen. Additionally the cooling module gives the required mass flow in case no heat exchanger is mounted to the engine. In this case the HPC exit temperature serves as cooling temperature. The result of the complete mass flow for cooling has an influence on the overall result being discussed in section 5.4. From table 5.5 it can be derived, that the reduction of required cooling mass flow when a heat exchanger is used is 41.75 % with respect to the default requirement. As the tool was used in a black box approach, the outputs cannot be further commented. As a concluding remark it can however be said that the required mass flow is overall quite low, but reflecting on the temperature difference that has to be achieved, the results are logical. The benefit of the heat exchanger can already be seen from this result.

The air pressure drop over both stages of the cooling blades is ∆p=712.79 Pa according to the cooling program used. Thus as compared with the pressure drop seen in ﬁgure 5.8, it is relatively small. Hence the pressure drop of the heat exchanger is higher than of the turbine cooling. 5.4.R ESULTSENGINEPERFORMANCEMODULE 75

Table 5.5: Cooling module results - mass ﬂows

Parameter Cooling mass ﬂow [kg/s]

m˙ cool,HE x single blade 0.0091 m˙ cool,HE x NGV1 0.7283 m˙ cool,HE x HPT1 1.3110 m˙ cool,HE x NGV2 0.7283 m˙ cool,HE x HPT2 1.3110 m˙ cool,HE x Total 4.0786

m˙ cool,T 3 single blade 0.0156 m˙ cool,T 3 NGV1 1.2503 m˙ cool,T 3 HPT1 2.2505 m˙ cool,T 3 NGV2 1.2503 m˙ cool,T 3 HPT2 2.2505 m˙ cool,T 3 Total 7.0017

Table 5.7: SFC comparison of engine with and without heat exchanger

Parameter Quantity SFC without HEx [g/kN.s] 7.929 SFC with HEx [g/kN.s] 7.8796 Difference in % 0.63

5.4. RESULTSENGINEPERFORMANCEMODULE

This section will give insight in the overall system performance of the GE-90-94B with a the adaptation of running on LNG and having the heat exchanger strapped on. The results are thus vicarious for the AHEAD engine. Running an off-design condition simulation of the engine produces the results shown in figure 5.10 and 5.11. As mentioned before the off-design condition is run for a bleed air sweep. The two conditions with and without heat exchanger equipped will be discussed in the following. In figure 5.10 the thrust versus bleed air can be seen. The graph shows a steady decrease in thrust with bleed air and a change in slope at bleed air m˙ 2.2kg/s. Without a heat exchanger mounted to the engine, the bleed air required is bleed = m˙ 4.0786kg/s. From this it can be determined that the thrust is F 399.55kN. Mounting the heat bleed = = exchanger on the system will reduce the bleed mass flow to m˙ 7.0017kg/s. This implies a thrust of bleed = F 368.63kN. The difference in thrust can be calculated to be 8.38 %. These results are summed up in table = 5.6.

Table 5.6: Thrust comparison of engine with and without heat exchanger

Parameter Quantity Thrust without HEx [kN] 368.63 Thrust with HEx [kN] 399.55 Difference in % 8.38

In figure 5.11 the specific fuel consumption for the engine with and without heat exchanger mounted to it can be seen. The graph indicates an increase in TSFC with an increasing bleed air. This was expected since the work needed to compress the air required for cooling is lost in a thermodynamic way. The TSFC without the heat exchanger mounted to the engine is TSFC 7.929g/kN.s. When the heat exchanger is used, the = specific fuel consumption reduces to TSFC 7.8796g/kN.s. The benefit gained from the heat exchanger is = rather low. In total an improvement of 0.68% is gained. These values are summed up in table 5.7 76 5.R ESULTS &DISCUSSION

Figure 5.10: Change in thrust with bleed air

Figure 5.11: Change in SFC with bleed air 5.5.S ENSITIVITY ANALYSIS 77

5.5. SENSITIVITY ANALYSIS In order to estimate the operational envelope of the heat exchanger it is necessary to perform an estimation of the operational limits. In the case of the heat exchanger this part was split in two. First of all the limits of the cross section are shown in ﬁgure 5.12. This design envelope ﬁts best to the program, since it does not include a length constraint.

Figure 5.12: Cross sectional operational envelope

This envelope guarantees the complete heat exchange to the design requirements stated in table 5.1. It can be seen in the graph, that the lower limit in terms of air mass flow is posed by the evaporation requirement. With air mass flows lower than this line no complete evaporation of LNG can be performed. Logically this line rises with an increase in LNG mass flow. On the other side of the operational envelope there are two lines closing the envelope. The red line indicates the limitation of heat transfer. This means that when the heat exchanger operates on this line, LNG will reach the inlet temperature of air by the end of the exchanger. it hence gives an absolute limit. The green line on the other side indicates the pressure drop limit for the heat exchanger. This pressure drop limit is determined from the pressure drop in the combustion chamber minus the pressure drop known from the cooling module for cooling the turbine. Next the design envelope for the fixed geometry heat exchanger will be given. This design envelope holds for the fixed heat exchanger length and diameters. Additionally the graph is only valid for the design temperatures and pressures. A deviation of those can alter this plot significantly in terms of shape of the boundaries and effectiveness lines. It can be retrieved from figure 5.13. To obtain this plot a double iteration had to be performed. Since for the design length not every flow condition will produce the outputs that were required as stated in table 5.1, some iterations were needed. At first the output temperature of the heat exchanger was iterated. This was done with a resolution of 5K in order to reach convergence within acceptable time limits. ± The mass flows then had to be adjusted in an iterative way until the required effectiveness for each LNG mass flow was reached. The accuracy requirement for this iteration was set to 1%. This was found to be a good compromise between accuracy and computation time. Hence the lines of constant effectiveness are accurate to 1%. It can be seen that the effectiveness is highest on the boundary of the LNG evaporation. This comes to no surprise when regarding the definition of effectiveness as given in relations 2.12 and 2.12. The temperature difference reached over the maximum temperature difference is a main criterion for effectiveness. The chart can be interpreted as follows. For a given LNG mass flow the effectiveness reduces with an increasing air mass flow. This is due to the fact, that the thermal capacity of LNG relative to the increasing air mass flow reduces. Thus, a higher mass flow cannot be cooled down as much as a lower mass flow can. Additionally 78 5.R ESULTS &DISCUSSION

Effectiveness plot for the heat exchanger 7 Evaporation limit Pressure drop limit 6 ǫ = 10%

5 ǫ = 20% 4 [kg/s] 3 Air ˙ m 2 ǫ = 30% Design condition 1 ǫ = 50%

0 0 0.2 0.4 0.6 0.8 1 m˙ LN G [kg/s]

Figure 5.13: Design envelope and effectivenesses the nonlinearity of the entire process with the change in phase contributes to the curved shape of the lines of constant effectiveness. When the air mass flow is high, the Reynolds number is high, resulting in a high Nus- selt number and hence a high heat transfer coefficient. For the single phase part this is beneficial, since the heat transfer is enhanced. For the two phase flow however it is a hurdle since it leads to a quick evaporation of LNG. Thus the length in which two phase flow is present is relatively short. Thus the benefit of having a high heat transfer coefficient is low. Due to this effect the temperature drop of air is mainly obtained by single phase region. This however takes longer due to the lower heat transfer coefficient or within the same length of heat exchanger a smaller temperature drop is reached. Hence the effectiveness is also lower. The graph shows, that the constant effectiveness has a peak value. This is due to the interplay of two effects. Firstly the above mentioned effect is given. However also the single phase part contributes to the overall performance. While on the one hand it is beneficial to have a conservative mass flow of air to obtain a long length of two phase flow, a conservative mass flow results in a longer length required for the single phase heat transfer. A compromise between these two effects has to be found in order to arrive at a satisfying result. Of course other criteria have can influence the choice of the design point. For a heat exchanger operating always under the same conditions, optimization is simple. The point of highest effectiveness will be chosen for the design. For the given application however it is also required, that the mass flow may be altered. This is due to the different thrust settings that occur during a flight. Since an optimized design is very sensitive to changes in mass flow, it is for the given application also better, not to choose such an optimal point. Overall the result reflects well the initial assumption regarding the expected heat exchanger effectiveness. The highest effectiveness for the heat exchanger was found at the operation conditions of m˙ 0.2kg/s and m˙ LNG = air = 0.16kg/s. Here the effectiveness is ² 74.02%. = 6

DESIGN SUGGESTION

Derived from the outcomes presented in chapter 5, a physical design suggestion can be made. This suggestion does not incorporate a mechanical analysis and therefore is purely based on the thermal outcome of the design. It was found that the pressure alone does not pose a severe criterion on the sheet thickness of the exchanger. It is however likely that the influence of temperature leads to the need of a higher sheet thickness. Thus an educated guess was made to come up with a suggested sheet thickness. This thickness was taken to be 3mm. Since the heat exchanger will have to be mounted to a gas turbine engine, size is an important criterion. For this reason the heat exchanger was folded several times to reduce its length. Also the modular build up allows for a curvature following the engine. In the given case however the heat exchanger was designed to be mounted on a flat surface. When the heat exchanger is designed, factors like temperature stress corrosion and fluid induced vibration can put constraints on the design shape and material used. For this reason, a material such as stainless steel is suggested as material. This material offers a high safety range and good overall properties with regard to heat transfer, stress resistance and price. With this material the weight of the heat exchanger will be M 158.13kg. = In figure 6.1 a render of the heat exchanger suggested can be found. The inlet diameter of the heat exchanger

Figure 6.1: Design suggestion for the heat exchanger was arbitrarily set to 0.1m. This size was chosen for the LNG stream as well as for the air stream. It ; =

79 80 6.D ESIGN SUGGESTION can be expected that in the entrance region of the heat exchanger the air stream will be highly nonuniform. Therefore the entrance part was elongated by L 10cm to allow the flow to develop. It is estimated that this = elongation however will not be sufficient to completely allow the flow to fully develop. A CFD study would be necessary to bring insight into the inlet behavior of the flow. This length addition was also required to attach inlet flanges to the exchanger. The flanges were chosen based on the criterion of standardized bolt diameters and manufacturability. Due to the bends of the exchanger manufacturing of the inner tubes becomes difficult. Two approaches could be used to solve this problem. The first one being an increase in diameter to allow for a fitting, the other would be to press-fit the tubes. If manufacturing of the tubes can be achieved as single piece, the shell can be easily built around the tubes. In order to hold the tubes in place, struts have to be inserted in the heat exchanger. The heat exchanger shells could easily be mounted to a structure by welding and bolting. A drawing of the exchanger can be found in appendix B. 7

CONCLUSIONS &RECOMMENDATIONS

This chapter will summarize the main ﬁndings of the thesis and discusses the overall insight gained into the topic. The chapter will furthermore answer the research question posed in chapter 1. First the main conclusion will be drawn while recommendations for future work will be given in section 7.2.

7.1. CONCLUSIONS

The question whether it is possible to assess the thermal performance of the cryogenic heat exchanger and its effects on the bleed air consumption of the cooling system of the gas turbine using semi empirical correlations can be answered now. The answer to the question is a conditional ’yes’. It was shown, that the thermal design of a heat exchanger using semi empirical methods can well be done as long as certain assumptions are made. The accuracy of the design is thus dependent on these assumptions. It is clear, that a detailed analysis of the fluid phenomena is not possible using these methods. However they provide a first design estimate, which later on can be verified using CFD code. This holds also for the cooling module. For the engine performance module the answer to the research question is case dependent. For this thesis the objective was to design a heat exchanger using sea level conditions. Since data from engines are usually provided for cruise condition, it is not possible to assess the engine performance correctly using analytical methods. For this reason the tool GSP was employed. This had to be done because the efficiencies of the turbomachinery changes depending on load condition. In real life this is approximated by compressor or turbine maps, which differ from engine to engine.

The main conclusions are thus:

• Care must be taken when operating at high pressures. If the pressure of the liquid is above the critical point no gasiﬁcation can be done. Instead the ﬂuid will enter a supercritical region.

• Implemented bends are not critical for the LNG side, but can signiﬁcantly increase the pressure drop on the shell side.

• The highest effectiveness of the heat exchanger is achieved at low mass flows close to the evaporation boundary. The lowest effectiveness can be found for high air mass flow at low LNG mass flow.

• The design of the heat exchanger is mainly driven by the requirements of pressure drop, LNG evaporation and temperature difference. The pressure drop being very low for the tested constellations is the upper limiting factor for the air mass flow. For LNG the pressure drop is not critical. It could be seen, that the highest heat exchanger effectivenesses is obtained for low mass flows for the suggested heat exchanger. This resulted in low flow speeds on the LNG side, resulting in a big overall share of two phase flow.

• High heat transfer capabilities on the shell side are beneficial to reduce the length of the single phase parts, but they also reduce the two phase parts. The change in geometry has influence on flow velocities as well as on the evaporation. Fins additionally introduce unilateral complexity in the design.

81 82 7.C ONCLUSIONS &RECOMMENDATIONS

• The results of this exchanger show, that the application of the heat exchanger can save up to 41.7 % of the normally required bleed air for this special case. This implies an increase in thrust of 30kN and small savings in the specific fuel consumption. Thus the heat exchanger is beneficial for the system. The overall benefit during aircraft operation however could not be determined.

7.2. RECOMMENDATIONS

This section will show up the recommendations for future works in this area. This section will discuss points to be addressed when working with the developed program, general advice for improving the methods used and points for special attention. The discussion will focus on the heat exchanger part only, since the two other parts were used in a black box approach. Since the heat exchanger is the part that was dealt with most and in most detail also the recommendations given in this section are the most detailed ones.

• Since no data existed beforehand, the results obtained by the program could not directly be validated. Thus, further research is needed in order to show that the design indeed complies with the requirements. At first it is suggested to perform a CFD analysis in order to obtain a impression on the flow distortion. It is expected that especially in the air entrance region substantial turbulence is induced. Furthermore the effect of fins throughout bents can introduce secondary flows that foster or deteriorate heat transfer and pressure drop. Finally it is suggested to build a prototype of the heat exchanger and perform real life measurements on the device. This is also necessary to determine the influence of the several assumptions made for the heat exchanger.

• A further point on which further studies could be developed is the macroscopic flow within the air compartments of the exchanger. Since the analyses of heat transfer were always based on the detailed and isolated flow conditions, the introduces error by this idealization could be of high influence. This holds for the heat transfer coefficient of the shell side tubes and the fins.

• If bends are made within the two phase region, further analysis has to be made. Currently bends are only incorporated within the single phase region

• The heat transfer coefficient of the two phase part should be validated against measurements. The comparison given in chapter 4 had to be made in order to come up with a quantitative estimate of the heat transfer coefficient. However in total it is unclear what the error induced by the semi empirical correlation is. Thus, the verification of it should be validated.

• The overall heat transfer coefficient can be calculated by several formulae. The one chosen in this con- text contains the thermal conductivity of the wall. This has a significant influence on the heat transfer. This comes from the fact, that the difference in heat transfer resistance of the fluids and the tube material is large. The error introduced by this might have significant impacts on the entire design.

• It was found out in the chapter on results, that the pressure drop overall is very low. It was said, that this is due to the very low flow speeds of both fluids. Therefore it would be interesting to implement a cross flow mode in the heat exchanger program to exploit also other flow configurations.

• The risk of entering a supercritical region is high when operating under high pressure. Then, a vaporization of LNG is no longer possible. This problem could be solved by reducing the pressure of the LNG stream and increase it again by a pump after the heat exchanger. Otherwise supercritical heat transfer offers a solution as well.

• Since the total length of the heat exchanger is relatively short, the ﬂow in the air compartment cannot be regarded as fully developed. This will imply a difference in the heat transfer abilities predicted. This should be implemented in future works with the code.

• Implementation of a better correlation for the multi-tube constellation is favored.

• As the superheat temperature is very high, it is likely that also ﬁlm boiling plays an important role in this heat transfer. An analysis on this would be recommended.

• Due to the double pipe heat exchanger assumption the temperature change along one section is overes- timated depending on the heat exchanger conﬁguration. The introduced error is thus zero for a double 7.2.R ECOMMENDATIONS 83

pipe conﬁguration and is highest for two tubes used. When the number of tubes increases the error reduces again.

BIBLIOGRAPHY

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TWOPHASEHEATTRANSFERCORRELATIONS

In this appendix the correlations used for verifying the two phase heat transfer coefﬁcients will be presented. In total three additional semi-empirical correlations were used. 1. Gungor-Winterton correlation 2. Bertsch correlation 3. Kandlikar correlation Since the Gungor-Winterton correlation is most similar to the Liu-Winterton relation it will be shown ﬁrst.

A.1. GUNGOR-WINTERTON The Gungor-Winterton correlation was presented in 1985 and printed in [51]. The relation contains data from veritcal as well as horizontal flow in tubes. The fluids used are water, refrigerants and ethylene glycol. Most cases deal with saturated boiling, but also subcooled boiling data is used. The method builds the two phase heat transfer coefficient out of coopers pool boiling coefficient paired with a suppression factor and the Dittus Boelter forced convection heat transfer coefficient combined with an enhancement factor. This is depicted in relation A.1. h Eh Sh (A.1) tp = l + pool Coopers pool boiling equation and the Dittus Boelter law are represented in formuale A.2 to A.3.

0.12¡ ¢ 0.55 0.5 0.67 h 55p log(p ) − M − q (A.2) pool = r − r

0.8 0.4 kt,l hl 0.023(Rel) (Prl ) (A.3) = Dh The factors E and S are calculated as depicted in relations A.4 and A.5.

µ 1 ¶0.86 E 24000Bo1.16 1.37 (A.4) = + Xtt 1 S (A.5) = 1 1.15 10 6E 2Re 1.17 + · − l In these relations Bo is the boiling number and Xtt the Martinelli parameter. The boiling number can be computed as depicted in reference A.6 while the Martinelli parameter is determined from equation A.7. q Bo (A.6) = λG µ ¶0.9 µ ¶0.5 µ ¶0.1 1 x˙ ρv µl Xtt − (A.7) = x˙ ρl µv

89 90 A.T WO PHASE HEAT TRANSFER CORRELATIONS

Gungor and Winterton additionally state, that in case a horizontal tube is used and the Froude number is less than 0.05 the factor E should be multiplied by E2 and S by S2 as given in relations A.8 and A.9. The Froude number can be computed according to reference A.10.

(0.1 2Fr ) E Fr − (A.8) 2 = S pFr (A.9) 2 = G2 Fr 2 (A.10) = ρl gd With this correlation ﬂow boiling in tubes as well as in annuli can be calculated.

A.2. BERTSCH

In addition to the heat transfer mechanisms of nucleate boiling and forced convection boiling Bertsch uses a confinement number to account for bubble confinement in small channels. Being based on 14 studies the hydraulic diameters of this method range between D 0.16mm to D 2.92mm. In general the sample h = h = points used cover a wide range in parameters. The correlation was published in 2008 and can be found in reference [55]. Also Bertsch gives the two phase heat transfer coefficient as a weighted sum of forced convection boiling and nucleate boiling. This can be seen in relation A.11.

h h S h F (A.11) FB = nb 2 + conv,tp The nucleate boiling will be determined by Cooper. This is depicted in relation A.12.

0.12 0.2 log10(RP )¡ ¢ 0.55 0.5 0.67 h 55p − · log(p ) − M − q (A.12) pool = r − r

When the roughness factors RP is set to 1 as suggested for unknown surface roughness, this relation reduces to A.2. The convective part of the heat transfer is modeled according to the phase. The general correlation can be seen in relation A.13. h h (1 x˙) h x˙ (A.13) conv,tp = conv,l · − + conv,v · The expression for hconv,l and hconv,v can be retrieved from A.14 with the according parameters for liquid and gas. This relation was retrieved from a laminar ﬂow correlation developed by Hausen.   D 0.0668 h Re Pr k  · L · ·  t,l hconv,l 3.66 2  (A.14) =  + h i 3  · Dh 1 0.04 Dh Re Pr + · L · · The Reynolds number is deﬁned as indicated in equation A.15 for liquid and A.16 for gas.

G Dh Rel · (A.15) = µl

G Dh Rev · (A.16) = µv The suppression factor is calculated by relation A.17, while the enhancement factor is given by A.18.

S 1 x˙ (A.17) 2 = − b Co 2 6 F 1 a e− · (x˙ x˙ ) (A.18) = + · · − The conﬁnement number reduced the enhancement factor. It is deﬁned as depicted in A.19. The value of σ represents the surface tension. Bertsch gives the database parameters a and b to be a 80 and b 0.6. = = s σ Co (A.19) = g (ρ ρ ) D 2 · l − v · h A.3.K ANDLIKAR 91

Table A.1: Coefﬁcients for the Kandlikar relation

Parameter Convective region Nucleate boiling region C1 1.1360 0.6683 C2 -0.9 -0.2 C3 667.2 1058.0 C4 0.7 0.7 C5 0.3 0.3

Table A.2: Fluid dependent parameter for the Kandlikar relation

Fluid F f l Water 1.00 R-11 1.30 R-12 1.50 R-13B1 1.31 R-22 2.20 R-113 1.30 R-114 1.24 R-152a 1.10 Nitrogen 4.70 Neon 3.50

A.3. KANDLIKAR Finally the Kandlikar relation is shown. In contrast to the other relations it is not build up on the convection heat transfer coefficient and the nucleate boiling coefficients. Those effects are accounted for by Kandlikar using the boiling and convection number. Additionally Kandlikar uses many constants in his main relation. Table A.1 gives the constants, while equation A.20 gives the main equation. In case of vertical flow or Fr lo > 0.04 the constant C5 is set to 0.

h h ¡C1CoC2(25Fr )C5 C3BoC4F ¢ (A.20) tp = l · lo + f l

The value of hl can be taken from A.3. The parameter Frlo represents the Froude number with all flow taken as liquid. It can be determined by A.10. Additionally F f l is a fluid dependent parameter, which can be taken from table A.2. Kandlikar calculates the two phase heat transfer coefficient twice with formula A.20. In the two calculations he uses one time the coefficients for convection and one time for nucleate boiling. He compared the two heat transfer coefficients and choses always the higher of the two to be the real two phase heat transfer coefficient.

DRAWINGOFTHEHEATEXCHANGER

93 Figure B.1: Test C

CODE

C.1. ACCPRESSURELOSS

Program/AccPressureloss.m 1%% Two phase flow subroutine 2 3%% Determine volume and initial parameters for two phase flow 4 Q=0.00002;%initial heat flux for evaporation inKJ 5 vol .sec=dx * r _ i ^2* pi;%sectional tube volume 6 M.lng=0.016828;%Molecular weight in kg/mol 7 Pc.lng=5322000;%Critical pressure data from NIST − 8 Ts=sat_temp(P_lng) ;%Saturation temperature 9 x_dot =0;%Initial vapor quality; assumed0 10 X_dotfinal(1)=x_dot;%Fill the endvector with first value 11 12%Determine mass per tube and shares of the section in kg 13 M_tot .sec=vol.sec * rho_lng_liq(T_lng,P_lng) ;%initial mass in one section 14 M_tot. tot=mdot_lng/No_tubes;%mass flow per tube 15 G=M_tot . tot /( r _ i ^2* pi);%Mass velocty per tube 16 P_lngdata_tp(1)=P_lngdata_sp1(end);%Fill the endvector with first value 17 18%% Main calculation 19 if type ==’Parallelflow’%If flow arrangement is parallel flow 20 for Counter=2:10000%Large value to create long loop (=while) 21 22%Determine Vapor quality 23 [x_dot, M_gas, M_tot,alfa ,delta_h_lg ,M_liquid,rho,rho_gas]=vapor_quality_tool(Q,x_dot,M_tot,T_lng, P_lng ) ; 24 25%Intermediate data saving 26 M_gas_total(Counter)=M_gas; 27 vapor_quality.lng(Counter)=x_dot; 28 alpha.lng(Counter)=alfa; 29 rho_lng_tp(Counter)=rho; 30 density_gas(Counter)=rho_gas; 31 32%Calculation of acclerational pressure drop 33 A=G^2*(x_dot^2/(alfa * rho_lng_gas(T_lng,P_lng))) ;%e x i t section 34 B=G^2 ((1 x_dot)^2/(rho_lng_liq(T_lng,P_lng) (1 a l f a ) ) ) ;%e x i t section * − * − 35 C=G^2 (vapor_quality .lng(Counter 1)^2/(alpha. lng(Counter 1) rho_lng_gas(T_lng , P_lngdata_tp(Counter * − − * 1) ) ) ) ;%entrance section − 36 D=G^2 (1 vapor_quality .lng(Counter 1) ) ^2/((1 alpha. lng(Counter 1) ) rho_lng_liq(T_lng ,P_lngdata_tp( * − − − − * Counter 1) ) ) ;%entrance section − 37 38 if isnan(B)%Case alpha and xdot =1 39 B=1/rho_lng_liq(T_lng,P_lng); 40 disp(’B is NaN AccPressureloss’) 41 end 42 if isnan(C)%Case alpha and xdot =0

95 96 C.C ODE

43 C=1/rho_lng_gas(T_lng,P_lngdata_tp(Counter 1) ) ; − 44 disp(’C is NaN AccPressureloss’) 45 end 46 47 delta_p_acc=A+B C D;%Accelerationa pressure drop − − 48 49%Intermediate data saving 50 delta_Pa.lng(Counter)=delta_p_acc; 51 Liqm.lng(Counter)=M_liquid.mass; 52 Gas(Counter)=M_gas; 53 54%Calculation of frictional pressure loss 55 [delta_p_fric ,Re_g,Re_l,A,B]=FricPressureloss(r_i ,G,x_dot,T_lng,P_lng,mdot_lng,No_tubes,dx); 56 57%Intermediate data saving 58 delta_Pf.lng(Counter)=delta_p_fric; 59 gaspressurelos(Counter)=B; 60 liqpressurelos(Counter)=A; 61 Relng_tp_gas(Counter)=Re_g; 62 Relng_tp_liq(Counter)=Re_l; 63 X_dotfinal(Counter)=x_dot; 64 delta_P_lng_tp(Counter)=delta_Pa.lng(Counter)+delta_Pf.lng(Counter);%Complete pressure drop on LNG side 65 66 67%Determine fluid properties 68 rho_a=rho_air(T_air,P_air); 69 my_a=my_air(T_air,P_air); 70 cp_a=cp_air(T_air,P_air); 71 k_t_a=kt_air(T_air,P_air); 72 Pr_lng=cp_lng_liq(T_lng,P_lng) * my_lng_liq(T_lng,P_lng)/kt_lng_liq(T_lng,P_lng) ;%Liquid Prandtl number 73 v_air=mdot_air / ( (pi ( r_O^2 No_tubes r_o^2))+(No_tubes No_ribs length_rib thickness_rib))/rho_a; * − * * * * 74 [Re_air,Pr_air]=imp_quan_shell(rho_a,v_air ,r_o,r_O,my_a,cp_a,k_t_a,No_tubes,No_ribs,length_rib , thickness_rib ,Ribs); 75 76%Calculate heat transfer parameters 77 [U,h_i,h_o,A_i,A_o,Re_air,Pr_air ,q,T_w,h_fin,T_fin,length_fin ,v_lng]= overall_heatcoeff_tp(Re_air, Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,No_tubes,X_dotfinal ,Pc,P_air ,T_air ,P_lng,T_lng, mdot_lng,Counter,M,Ribs ,No_ribs , thickness_rib ,length_rib ,No_rings ,Pr_lng ,T_w, v_air) ; 78 79%Calculate the energy going into the next section for evaporization 80 Q=q* r _ i *2* pi *dx/(v_lng/dx)/1000;%Energy in Kilojoules that goes into the liquid mass in the section 81 82%Intermediate data saving 83 q_tp ( Counter )=Q; 84 q_tp2 ( Counter )=q ; 85 U_tp( Counter )=U; 86 Fin_h_tp(Counter)=h_fin; 87 h_i_tp(Counter)=h_i; 88 h_o_tp(Counter)=h_o; 89 Re_air_tp(Counter)=Re_air; 90 Pr_air_tp(Counter)=Pr_air; 91 Pr_lng_tp(Counter)=Pr_lng; 92 Walltemp_tp(Counter)=T_w; 93 Fin_T_profile_tp(:,Counter)=T_fin; 94 v_lng_tp(Counter)=v_lng; 95 v_air_tp(Counter)=v_air; 96 rho_air_tp(Counter)=rho_a; 97 cp_air_tot_tp(Counter)=cp_a; 98 cp_lng_tot_tp(Counter)=cp_lng_liq(T_lng,P_lng); 99 100%Compute new temperatures 101 deltaT_air_tp=U No_tubes/(mdot_air cp_air(T_air ,P_air) dx ) ( T_air T_lng ) dx ;%Temperature * * * * − * difference of Air in Kelvin 102 T_air=T_air deltaT_air_tp;%New temperature of Air − 103 104%Intermediate data saving 105 T_air_tp(Counter)=T_air; 106 T_lng_tp(Counter)=T_lng; C.1.A CCPRESSURELOSS 97

107 deltaTair_tp(Counter)=deltaT_air_tp; 108 deltaTLNG_tp(Counter)=0; 109 110%Pressure drop calculations 111 P_lng=P_lng (delta_p_acc+delta_p_fric)/100000; − 112 [P_air ,delta_p_air]=Newpressure_air(P_air ,Re_air,mdot_air,dx,r_o,r_O,rho_air_tp ,No_tubes,Ribs, No_ribs,thickness_rib ,length_rib ,Counter) ; 113 P_air=P_air delta_p_air/100000; − 114 115%Intermediate data saving 116 P_lngdata_tp(Counter)=P_lng; 117 delta_P_air_tp(Counter)=delta_p_air; 118 P_airdata_tp(Counter)=P_air; 119 120 if x_dot >= 1%break criterion vaporization 121 break 122 end 123 124 125 end 126%% Appending singelphase and twophase flows 127 128 Length=1:Counter ;%Lenght count of the exchanger 129 Length=Length*dx ;%calculate the acutal length 130 temp=Length+co (end);%Add single phase length to two phase length 131 Total_Length = [co temp];%create total length vector 132 133%% Filling last positions of data, clear data and append variables 134 P_lng_total = [P_lngdata_sp1,P_lngdata_tp]; 135 P_airdata_tp(1)=P_airdata_sp1(end); 136 P_air_total = [P_airdata_sp1, P_airdata_tp]; 137 T_lng_tp(1)=T_lng_sp1(end); 138 T_lng_total = [T_lng_sp1 T_lng_tp]; 139 T_air_tp(1)=T_air_sp1(end); 140 T_air_total = [T_air_sp1 T_air_tp]; 141 U_tp (1)=U_sp1(end); 142 U_total = [U_sp1 U_tp ] ; 143 h_i_tp(1)=h_i_sp1(end); 144 h_i_tp(2)=h_i_tp(3); 145 h_i_total = [h_i_sp1 h_i_tp]; 146 h_o_tp(1)=h_o_sp1(end); 147 h_o_total = [h_o_sp1 h_o_tp]; 148 X_dot_total = [quality vapor_quality.lng]; 149 deltaTair_tp(1)=deltaTair_sp1(end); 150 deltaTLNG_tp(1)=deltaTLNG_sp1(end); 151 deltaT_air_total = [deltaTair_sp1 deltaTair_tp]; 152 deltaT_lng_total = [deltaTLNG_sp1 deltaTLNG_tp]; 153 Re_air_tp(1) = Reair_sp1(end); 154 Pr_air_tp(1) = Prair_sp1(end); 155 Reynolds_air_total = [Reair_sp1 Re_air_tp]; 156 Prandtl_air_total = [Prair_sp1 Pr_air_tp]; 157 Walltemp_tp(1) = Walltemp_sp1(end); 158 Walltemp_sptp = [Walltemp_sp1 Walltemp_tp]; 159 Fin_h_tp(1) = Fin_h_sp1(end); 160 Total_h_fin = [Fin_h_sp1 Fin_h_tp]; 161 Fin_T_profile_tp(:,1)=Fin_T_profile_tp(:,2); 162 Total_fin_t = [Fin_T_profile_sp1 Fin_T_profile_tp]; 163 delta_P_air_tp(1)=delta_P_air_sp1(end); 164 delta_P_lng_tp(1)=delta_P_lng_sp1(end); 165 delta_P_tot_lng=[delta_P_lng_sp1 delta_P_lng_tp]; 166 delta_P_tot_air=[delta_P_air_sp1 delta_P_air_tp]; 167 v_air_tp(1)=v_air_tp(2); 168 v_lng_tp(1)=v_lng_tp(2); 169 v_lng_tot=[v_lng_sp1 v_lng_tp]; 170 v_air_tot=[v_air_sp1 v_air_tp]; 171 density_gas(1)=density_gas(2); 172 rho_lng_tp(1)=rho_lng_sp1(end); 173 rho_air_tp(1)=rho_air_sp1(end); 174 rho_air_tot=[rho_air_sp1 rho_air_tp]; 175 rho_lng_tot=[rho_lng_sp1 rho_lng_tp]; 176 Relng_tp_gas(1)=Relng_tp_gas(2); 98 C.C ODE

177 Relng_tp_liq(1)=Relng_tp_liq(2); 178 Pr_lng_tp(1)=Pr_lng_tp(2); 179 180 elseif type ==’Counter flow’ − 181 for Counter=2:10000%Large value to create long loop (=while) 182 183%Determine Vapor quality 184 [x_dot, M_gas, M_tot,alfa ,delta_h_lg ,M_liquid,rho,rho_gas]=vapor_quality_tool(Q,x_dot,M_tot,T_lng, P_lng ) ; 185 186%Storing data 187 M_gas_total(Counter)=M_gas; 188 vapor_quality.lng(Counter)=x_dot; 189 alpha.lng(Counter)=alfa; 190 rho_lng_tp(Counter)=rho; 191 density_gas(Counter)=rho_gas; 192 193%Calculation of acclerational pressure loss 194 A=G^2*(x_dot^2/(alfa * rho_lng_gas(T_lng,P_lng))) ;%e x i t section 195 B=G^2 ((1 x_dot)^2/(rho_lng_liq(T_lng,P_lng) (1 a l f a ) ) ) ;%exit section * − * − 196 C=G^2 (vapor_quality .lng(Counter 1)^2/(alpha. lng(Counter 1) rho_lng_gas(T_lng , P_lngdata_tp(Counter * − − * 1) ) ) ) ;%entrance section − 197 D=G^2 (1 vapor_quality .lng(Counter 1) ) ^2/((1 alpha. lng(Counter 1) ) rho_lng_liq(T_lng ,P_lngdata_tp( * − − − − * Counter 1) ) ) ;%entrance section − 198 199 if isnan(B)%Case alpha and xdot =1 200 B=1/rho_lng_liq(T_lng,P_lng); 201 disp(’B is NaN AccPressureloss’) 202 end 203 if isnan(C)%Case alpha and xdot =1 204 C=1/rho_lng_gas(T_lng,P_lngdata_tp(Counter 1) ) ; − 205 disp(’C is NaN AccPressureloss’) 206 end 207 208 delta_p_acc=A+B C D;%Accelerationa pressure drop − − 209 210%Intermediate data saving 211 delta_Pa.lng(Counter)=delta_p_acc; 212 Liqm.lng(Counter)=M_liquid.mass; 213 Gas(Counter)=M_gas; 214 215%Calculation of frictional pressure loss 216 [delta_p_fric ,Re_g,Re_l,A,B]=FricPressureloss(r_i ,G,x_dot,T_lng,P_lng,mdot_lng,No_tubes,dx); 217 218%Intermediate data saving 219 delta_Pf.lng(Counter)=delta_p_fric; 220 gaspressurelos(Counter)=B; 221 liqpressurelos(Counter)=A; 222 Relng_tp_gas(Counter)=Re_g; 223 Relng_tp_liq(Counter)=Re_l; 224 X_dotfinal(Counter)=x_dot; 225 delta_P_lng_tp(Counter)=delta_Pa.lng(Counter)+delta_Pf.lng(Counter);%Complete pressure drop on LNG side 226 227 228 229%Determine fluid properties 230 rho_a=rho_air(T_air,P_air); 231 my_a=my_air(T_air,P_air); 232 cp_a=cp_air(T_air,P_air); 233 k_t_a=kt_air(T_air,P_air); 234 v_air=mdot_air / ( (pi ( r_O^2 No_tubes r_o^2))+(No_tubes No_ribs length_rib thickness_rib))/rho_a; * − * * * * 235 Pr_lng=cp_lng_liq(T_lng,P_lng) * my_lng_liq(T_lng,P_lng)/kt_lng_liq(T_lng,P_lng) ;%Liquid Prandtl number 236 [Re_air,Pr_air]=imp_quan_shell(rho_a,v_air ,r_o,r_O,my_a,cp_a,k_t_a,No_tubes,No_ribs,length_rib , thickness_rib ,Ribs); 237 238%Calculate heat transfer parameters 239 [U,h_i,h_o,A_i,A_o,Re_air,Pr_air ,q,T_w,h_fin,T_fin,length_fin ,v_lng]= overall_heatcoeff_tp(Re_air, Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,No_tubes,X_dotfinal ,Pc,P_air ,T_air ,P_lng,T_lng, mdot_lng,Counter,M,Ribs ,No_ribs , thickness_rib ,length_rib ,No_rings ,Pr_lng ,T_w, v_air) ; C.1.A CCPRESSURELOSS 99

240 241%Calculate the energy going into the next section for evaporization 242 Q=q* r _ i *2* pi *dx/(v_lng/dx)/1000;%Energy in Kilojoules that goes into the liquid mass in the section 243 244%Intermediate data saving 245 q_tp ( Counter )=Q; 246 q_tp2 ( Counter )=q ; 247 U_tp( Counter )=U; 248 Fin_h_tp(Counter)=h_fin; 249 h_i_tp(Counter)=h_i; 250 h_o_tp(Counter)=h_o; 251 Re_air_tp(Counter)=Re_air; 252 Pr_air_tp(Counter)=Pr_air; 253 Pr_lng_tp(Counter)=Pr_lng; 254 Walltemp_tp(Counter)=T_w; 255 Fin_T_profile_tp(:,Counter)=T_fin; 256 v_lng_tp(Counter)=v_lng; 257 v_air_tp(Counter)=v_air; 258 rho_air_tp(Counter)=rho_a; 259 cp_air_tot_tp(Counter)=cp_a; 260 cp_lng_tot_tp(Counter)=cp_lng_liq(T_lng,P_lng); 261 262%Compute new temperatures 263 deltaT_air_tp=U No_tubes/(mdot_air cp_air(T_air ,P_air) dx ) ( T_air T_lng ) dx ; * * * * − * 264 T_air=T_air+deltaT_air_tp; 265 266%Intermediate data saving 267 T_air_tp(Counter)=T_air; 268 T_lng_tp(Counter)=T_lng; 269 deltaTair_tp(Counter)=deltaT_air_tp; 270 deltaTLNG_tp(Counter)=0; 271 272%Pressure drop calculations 273 P_lng=P_lng (delta_p_acc+delta_p_fric)/100000; − 274 [P_air ,delta_p_air]=Newpressure_air(P_air ,Re_air,mdot_air,dx,r_o,r_O,rho_air_tp ,No_tubes,Ribs, No_ribs,thickness_rib ,length_rib ,Counter) ; 275 P_air=P_air delta_p_air/100000; − 276 277%Intermediate data saving 278 P_lngdata_tp(Counter)=P_lng; 279 delta_P_air_tp(Counter)=delta_p_air; 280 P_airdata_tp(Counter)=P_air; 281 282 if x_dot >= 1%break criterion vaporization 283 break 284 end 285 286 287 end 288%% Appending singelphase and twophase flows 289 290 Length=1:Counter ;%Lenght count of the exchanger 291 Length=Length*dx ;%calculate the acutal length 292 temp=Length+co (end);%Add single phase length to two phase length 293 Total_Length = [co temp];%create total length vector 294 295%% Filling last positions of data, clear data and append variables 296 P_lng_total = [P_lngdata_sp1,P_lngdata_tp]; 297 P_airdata_tp(1)=P_airdata_sp1(end); 298 P_air_total = [P_airdata_sp1, P_airdata_tp]; 299 T_lng_tp(1)=T_lng_sp1(end); 300 T_lng_total = [T_lng_sp1 T_lng_tp]; 301 T_air_tp(1)=T_air_sp1(end); 302 T_air_total = [T_air_sp1 T_air_tp]; 303 U_tp (1)=U_sp1(end); 304 U_total = [U_sp1 U_tp ] ; 305 h_i_tp(1)=h_i_sp1(end); 306 h_i_tp(2)=h_i_tp(3);%remove peak at transition from sp to tp 307 h_i_total = [h_i_sp1 h_i_tp]; 308 h_o_tp(1)=h_o_sp1(end); 100 C.C ODE

309 h_o_total = [h_o_sp1 h_o_tp]; 310 X_dot_total = [quality vapor_quality.lng]; 311 deltaTair_tp(1)=deltaTair_sp1(end); 312 deltaTLNG_tp(1)=deltaTLNG_sp1(end); 313 deltaT_air_total = [deltaTair_sp1 deltaTair_tp]; 314 deltaT_lng_total = [deltaTLNG_sp1 deltaTLNG_tp]; 315 Re_air_tp(1) = Reair_sp1(end); 316 Pr_air_tp(1) = Prair_sp1(end); 317 Reynolds_air_total = [Reair_sp1 Re_air_tp]; 318 Prandtl_air_total = [Prair_sp1 Pr_air_tp]; 319 Walltemp_tp(1) = Walltemp_sp1(end); 320 Walltemp_sptp = [Walltemp_sp1 Walltemp_tp]; 321 Fin_h_tp(1) = Fin_h_sp1(end); 322 Total_h_fin = [Fin_h_sp1 Fin_h_tp]; 323 Fin_T_profile_tp(:,1)=Fin_T_profile_tp(:,2);%removing the first0 324 Total_fin_t = [Fin_T_profile_sp1 Fin_T_profile_tp]; 325 delta_P_air_tp(1)=delta_P_air_sp1(end);%filling the first position with last position of single phase flow 326 delta_P_lng_tp(1)=delta_P_lng_sp1(end);%filling the first position with last position of single phase flow 327 delta_P_tot_lng=[delta_P_lng_sp1 delta_P_lng_tp]; 328 delta_P_tot_air=[delta_P_air_sp1 delta_P_air_tp]; 329 v_air_tp(1)=v_air_tp(2); 330 v_lng_tp(1)=v_lng_tp(2); 331 v_lng_tot=[v_lng_sp1 v_lng_tp]; 332 v_air_tot=[v_air_sp1 v_air_tp]; 333 density_gas(1)=density_gas(2); 334 rho_lng_tp(1)=rho_lng_sp1(end); 335 rho_air_tp(1)=rho_air_sp1(end); 336 rho_air_tot=[rho_air_sp1 rho_air_tp]; 337 rho_lng_tot=[rho_lng_sp1 rho_lng_tp]; 338 Relng_tp_gas(1)=Relng_tp_gas(2); 339 Relng_tp_liq(1)=Relng_tp_liq(2); 340 Pr_lng_tp(1)=Pr_lng_tp(2); 341 342 end 343 344%% Prepare plots for case if temperature is reached before phase change is complete 345 346 if type ==’Counter flow’%If flow arrangement is counter flow − 347 if T_air >= T_air_in 348 P_final_lng=P_lng_total; 349 P_final_air = P_air_total; 350 T_final_lng = T_lng_total; 351 T_final_air = T_air_total; 352 U_final = U_total ; 353 h_i_final = h_i_total; 354 h_o_final = h_o_total; 355 X_dot_final = X_dot_total; 356 Reynolds_air_final = Reynolds_air_total; 357 Prandtl_air_final = Prandtl_air_total; 358 Total_Walltemp = Walltemp_sptp; 359 else spphasetwo 360 end 361 elseif type ==’Parallelflow’%If flow arrangement is parallel flow 362 if T_air <= T_req 363 P_final_lng=P_lng_total; 364 P_final_air = P_air_total; 365 T_final_lng = T_lng_total; 366 T_final_air = T_air_total; 367 U_final = U_total ; 368 h_i_final = h_i_total; 369 h_o_final = h_o_total; 370 X_dot_final = X_dot_total; 371 Reynolds_air_final = Reynolds_air_total; 372 Prandtl_air_final = Prandtl_air_total; 373 Total_Walltemp = Walltemp_sptp; 374 else spphasetwo 375 end 376 end C.2. BEND_PRESSUREDROP 101

C.2. BEND_PRESSUREDROP

Program/bend_pressuredrop.m 1 function [delta_P_bend] = bend_pressuredrop (L,rho_in,v_in ,D_h,type,loc) 2%function to determine the pressure drop due toa 180\deg bend of the heat 3%exchanger. Some parameters are fixed, like the roughness of the pipe and 4%the dependency of Reynoldsnumber to the given case. Due to the inaccurate 5%way of factor prediction by means of graphs, the Reynoldsnumbers were 6%defined as follows: Re_air=2.1e4, Re_lng_bpc=4.5e3 Re_lng_apc=4.5e4 7 8%INPUTS: length of the bend, entrance density, inlet flow speed, hydraulic 9%diameter, type of fluid, location of fluid 10%OUTPUTS: pressure drop due to bend 11 12%% Determine case according to the graphs of Shah 13%e=0.025;%typical surface roughness for steel pipe, values of roughness 14%according to Shah 15 if type ==’Air’ 16 Kb_star =1; 17 C_Re=Kb_star/(Kb_star 0.2 2+0.2) ; − * 18 f_rough =0.0065; 19 f_smooth =0.006; 20 elseif type ==’LNG’ 21 if loc ==’bpc’%location of bend:’bpc’=before phase change 22 Kb_star =0.3; 23 C_Re=2.2; 24 f_rough =0.011; 25 f_smooth =0.01; 26 elseif loc ==’apc’%location of bend:’apc’=after phase change 27 Kb_star =0.3; 28 C_Re=1.5; 29 f_rough =0.0075; 30 f_smooth=0.0055; 31 end 32 end 33 C_dev=1;%correction factor for bends bigger than 100\deg 34 C_rough=f_rough/f_smooth; 35 Kb=Kb_star *C_Re*C_dev*C_rough ; 36 f=f_rough ; 37 Kb_t=Kb+f *4*L/D_h; 38 delta_P_bend=Kb_t* rho_in * v_in ^2/2;

C.3. CLEARING

Program/clearing.m 1%% Clearing of variables for iteration 2 3 clear’A’’A_i’’A_o’’B’’C’’Counter’’D’’Fin_T_profile_sp1’’Fin_T_profile_sp2’’ Fin_T_profile_tp’’Fin_h_sp1’’Fin_h_sp2’’Fin_h_tp’’G’’Gas’’L’’Liqm’’M’’M_gas’’ M_gas_total’’M_liquid’’M_tot’’P_air’’P_air_total’’P_airdata_sp1’’P_airdata_sp2’’ P_airdata_tp’’P_lng’’P_lng_total’’P_lngdata_sp1’’P_lngdata_sp2’’P_lngdata_tp’’Pc’’ Pr_air’’Pr_air_tp’’Pr_lng’’Prair_sp1’’Prair_sp2’’Prandtl_air_final’’Prandtl_air_total’’ Prlng_sp1’’Prlng_sp2’’Q’’Re_air’’Re_air_tp’’Re_g’’Re_l’’Re_lng’’Reair_sp1’’ Reair_sp2’’Relng_sp1’’Relng_sp2’’Reynolds_air_final’’Reynolds_air_total’’T_air’’T_air_sp1 ’’T_air_sp2’’T_air_total’’T_air_tp’’T_fin’’T_final_air’’T_final_lng’’T_lng’’ T_lng_sp1’’T_lng_sp2’’T_lng_total’’T_lng_tp’’T_w’’Total_Length’’Total_Walltemp’ 4 clear’Total_fin_t’’Total_h_fin’’Ts’’U’’U_final’’U_sp1’’U_sp2’’U_total’’U_tp’’ Walltemp_sp1’’Walltemp_sp2’’Walltemp_sptp’’Walltemp_tp’’X_dot_final’’X_dot_total’’ X_dotfinal’’alfa’’alpha’’ans’’arc’’cenmid’’censhell’’centers_rings’’co’’co2’’ converged1’’converged2’’counter’’counter2’’cp_a’’cp_air_out’’cp_lng_out’’cp_lng_tot_sp1 ’’cp_lng_tot_sp2’’deltaTLNG_sp1’’deltaTLNG_sp2’’deltaTLNG_tp’’deltaT_air_sp’’ deltaT_air_sp2’’deltaT_air_total’’deltaT_air_tp’’deltaT_lng_sp’’deltaT_lng_sp2’’ deltaT_lng_total’’deltaTair_sp1’’deltaTair_sp2’’deltaTair_tp’’deltaTlng_sp2’’ delta_P_air_sp1’’delta_P_air_sp2’’delta_P_air_tp’’delta_P_lng_sp1’’delta_P_lng_sp2’’ delta_P_lng_tp’’delta_P_tot_air’’delta_P_tot_lng’’delta_Pa’’delta_Pf’’delta_h_lg’’ delta_p_acc’’delta_p_air’ 5 clear’delta_p_fric’’delta_p_lng’’density_gas’’gaspressurelos’’h_fin’’h_i’’h_i_final’’ h_i_sp1’’h_i_sp2’’h_i_total’’h_i_tp’’h_o’’h_o_final’’h_o_sp1’’h_o_sp2’’h_o_total’’ 102 C.C ODE

h_o_tp’’i’’k_t_a’’k_t_air’’k_t_lng’’ktlng_sp1’’ktlng_sp2’’liqpressurelos’’mdot1’’ mdot2’’mdot_air’’mdot_corr’’my_a’’my_air_out’’my_lng_out’’q’’q_tp’’quality’’ radial_distance’’radii’’radii_rings’’radius’’radmid’’radshell’’rho’’rho_a’’ rho_air_out’’rho_air_sp1’’rho_air_sp2’’rho_air_tot’’rho_air_tp’’rho_gas’’rho_lng_out’’ rho_lng_sp1’’rho_lng_sp2’’rho_lng_tot’’rho_lng_tp’’ring_arc’’ring_circ’’ring_spacing’’ temp’’v_air’’v_air_sp1’’v_air_sp2’’v_air_tot’ 6 clear’v_air_tp’’v_lng’’v_lng_sp1’’v_lng_sp2’’v_lng_tot’’v_lng_tp’’vapor_quality’’ visc_lng_sp1’’visc_lng_sp2’’vol’’x_dot’

C.4. DARCY_F

Program/darcy_f.m 1%Function to determine the Darcy friction factor in dependence from the 2%Reynoldsnumber 3%INPUT: Reynolds number 4%OUTPUT: friction factor 5 function f = darcy_f(Re) 6 f =(0.79 log(Re) 1.64)^ 2; * − − 7 end

C.5. FIN_TEMP_DISTR

Program/ﬁn_temp_distr.m 1%Function file to determine the temperature profile alonga fin 2%INPUTS:heat transfer coefficient fin, thermal condutivity of tube 3%material, thickness of rib, wall temperature, air temperature, length of 4%rib 5%OUTPUTS: Temperature and length points 6 function[T,length]=fin_temp_distr(h_fin ,k_t_tube ,thickness_rib ,T_w,T_air ,length_rib) 7 m=sqrt(2 * h_fin/k_t_tube/thickness_rib) ; 8 theta_0=T_w T_air ; − 9 10%Calculated constants from boundary conditions for adiabatic tip d_theta/dx=0 11 c2=theta_0 *1/(exp(2 *m* length_rib)+1); 12 c1=theta_0 c2 ; − 13 co=0; 14 for i=0:0.00001:length_rib%resolution along the rib 15 co=co+1; 16 length(co)=i; 17 theta ( co )=c1 exp( m i )+c2 exp (m i ) ;%excess temperature * − * * * 18 end 19 T=theta+T_air ;%real temperature of fin 20 end

C.6. FLATRATEPRESSURE

Program/ﬂatratepressure.m 1%Function to determine the pressure drop due to entrance and exit losses, 2%In case of exit losses the factor Kc should be replaced by Ke 3 4%INPUTS: entrance density, entrance velocity, exit velocity, Kc 5%OUTPUTS: pressure drop 6 function [Delta_P_flatrate] = flatratepressure(rho_in,v1,v2,Kc) 7 8 deltaP1=rho_in v2^2/2 (1 (v1/v2)^2);%sudden contraction * * − 9 deltaP2=Kc* rho_in *v2^2/2;%irreversible expansion 10 Delta_P_flatrate=deltaP1+deltaP2;

C.7. FRICPRESSURELOSS

Program/FricPressureloss.m 1% Function to determine the frictional pressure drop along the two phase 2% flow 3% Basic approach: The Reynoldsnumber is calculated according to the mass C.8. HEATOFVAP 103

4% flow and diameter of one tube. The viscosity of the mixture is 5% calculated with an weighted mass fractional average for gas and liquid 6% separately. The pressure drop of each flow is calculated separately with 7% the respective mass flow of the substance under the assumption of having 8% the shared Reynoldsnumber of the mixed flow. 9 10%INPUTS: inner radius of tube, mass velocity, vapor quality, Temperature, 11%pressure(, mass flow, number of tubes), step length 12%OUTPUTS: frictional pressure drop, gas reynolds number, liquid reynolds 13%number,A,B 14 function[delta_p_fric ,Re_g,Re_l ,A,B]=FricPressureloss(r_i ,G,x_dot ,T,P,mdot_lng,No_tubes,dx) 15 16%diameter 17 d=2* r _ i ; 18 19%Reynolds numbers 20 Re_g=G*d/my_lng_gas(T,P) ; 21 Re_l=G*d/my_lng_liq(T,P) ; 22 23%Pressure drop calculation 24 A=tp_fric_fac(Re_l) *G^2/(2* rho_lng_liq(T,P) *d) ; 25 B=tp_fric_fac(Re_g) *G^2/(2* rho_lng_gas(T,P) *d) ; 26 GG=A+2 (B A) x_dot ; * − * 27 delta_p_fric =(GG (1 x_dot)^(1/3)+B x_dot^3) dx ; * − * *

C.8. HEATOFVAP

Program/heatofvap.m 1%Latent heat of vaporization function 2%Although dependent on both pressure and temperature, it is assumed, since 3%the boiling line consists out of single points of temperature and 4%pressure, that the boiling temperature put into the file is equivalent to 5%its boiling pressure. This is calculated in another file. 6%Data inJ/kg 7 8%INPUTS: Temperature 9%OUTPUTS: latent heat of vaporization inKJ/kg 10 11 function delta_h_lg = heatofvap(T) 12 methane=[165.87 343560 13 167.18 336510 14 168.44 329440 15 169.66 322320 16 170.84 315150 17 171.99 307920 18 173.11 300600 19 174.19 293180 20 175.25 285660 21 176.27 278000 22 177.27 270190 23 178.25 262210 24 179.21 254030 25 180.14 245630 26 181.05 236960 27 181.94 227980 28 182.81 218650 29 183.66 208890 30 184.49 198610 31 185.31 187710 32 186.11 176030 33 186.89 163320 34 187.66 149210 35 188.41 133080 36 189.15 113610 37 189.87 87238]; 38 delta_h_meth=qinterp1(methane(: ,1) ,methane(: ,2) ,T) ; 39 40 ethane=[184.33 489720 41 198.19 470550 104 C.C ODE

42 207.42 456550 43 214.54 444920 44 220.43 434700 45 225.51 425400 46 229.99 416780 47 234.03 408650 48 237.71 400900 49 241.1 393460 50 244.25 386260 51 247.2 379250 52 249.98 372410 53 252.6 365700 54 255.09 359090 55 257.47 352570 56 259.74 346120 57 261.91 339720 58 264 333360 59 266.01 327030 60 267.95 320700 61 269.82 314380 62 271.63 308050 63 273.38 301700 64 275.08 295320 65 276.73 288900 66 278.34 282430 67 279.9 275890 68 281.42 269280 69 282.9 262580 70 284.34 255770 71 285.75 248850 72 287.13 241790 73 288.48 234570 74 289.79 227170 75 291.08 219560 76 292.34 211710 77 293.57 203570 78 294.78 195110 79 295.96 186240 80 297.12 176890 81 298.26 166950 82 299.37 156260 83 300.46 144570 84 301.53 131500 85 302.58 116380 86 303.61 97737 87 304.61 70913]; 88 delta_h_eth=qinterp1(ethane(: ,1) ,ethane(: ,2) ,T); 89 90 delta_h_lg=(delta_h_meth *0.9+delta_h_eth * 0 . 1 ) /1000;%Output in kJ/kg

C.9. HEX_RUN

Program/hex_run.m 1%% First single phase flow subroutine 2 3%% Pre f i l l vectors with initial inputs − 4 if type ==’Parallelflow’%If flow arrangement is parallel flow 5 T_air=T_air_in;%Set initial Temperature Air 6 T_lng=T_lng_in;%Set initial Temperature LNG 7 T_air_sp1(1)=T_air;%Fill the endvector with first value 8 T_lng_sp1(1)=T_lng;%Fill the endvector with first value 9 P_airdata_sp1(1)=P_air_in;%Fill the endvector with first value 10 P_lngdata_sp1(1)=P_lng_in;%Fill the endvector with first value 11 P_lng=P_lng_in ;%Set initial Pressure LNG 12 P_air=P_air_in ;%Set initial Pressure Air 13 14 elseif type ==’Counter flow’%If flow arrangement is counter flow − 15 T_air=T_air_ass ;%Set initial Temperature Air 16 T_lng=T_lng_in ;%Set initial Temperature LNG C.9. HEX_RUN 105

17 T_air_sp1(1)=T_air;%Fill the endvector with first value 18 T_lng_sp1(1)=T_lng;%Fill the endvector with first value 19 P_airdata_sp1(1)=P_air_in;%Fill the endvector with first value 20 P_lngdata_sp1(1)=P_lng_in;%Fill the endvector with first value 21 P_lng=P_lng_in ;%Set initial Pressure LNG 22 P_air=P_air_in ;%Set initial Pressure Air 23 end 24 25%% Pressure drop calculations due to entrance effects 26 27%LNG entrance into head 28 v1_lng=mdot_lng /(pi * r_entrance^2)/rho_lng(T_lng ,P_lng) ;%initial flow speed 29 v2_lng=mdot_lng /(pi *r_O^2*No_tubes)/rho_lng(T_lng ,P_lng) ;%second flow speed 30 delta_P_flatrate_lng1 = flatratepressure(rho_lng(T_lng,P_lng) ,v1_lng,v2_lng,Kc_lng);%pressure drop 31 P_lng=P_lng delta_P_flatrate_lng1/100000;%set new pressure − 32 33%LNG entrance into tubes 34 v1_lng=mdot_lng /(pi *r_O^2)/rho_lng(T_lng ,P_lng) ;%initial flow speed 35 v2_lng=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng(T_lng ,P_lng) ;%second flow speed 36 delta_P_flatrate_lng2 = flatratepressure(rho_lng(T_lng,P_lng) ,v1_lng,v2_lng,Kc_lng);%pressure drop 37 P_lng=P_lng delta_P_flatrate_lng2/100000;%set new pressure − 38 39%Air entrance into hex 40 v1_air=mdot_air /(pi * r_entrance^2)/rho_air(T_air ,P_air) ;%initial flow speed 41 v2_air=mdot_air /(pi ( r_O^2 r _ i ^2 No_tubes))/rho_air(T_air ,P_air) ;%second flow speed * − * 42 delta_P_flatrate_air = flatratepressure(rho_air(T_air,P_air) ,v1_air ,v2_air ,Kc_air);%pressure drop 43 P_air=P_air delta_P_flatrate_air/100000;%set new pressure − 44 45 46 rho_air_sp1(1)=rho_air(T_air ,P_air);%Fill the endvector with first value 47 rho_lng_sp1(1)=rho_lng(T_lng,P_lng) ;%Fill the endvector with first value 48 49%% Main Calculations 50 if type ==’Parallelflow’%If flow arrangement is parallel flow 51 for i=1:100000%Large value to create long loop (=while) 52%Determine fluid properties 53 [k_t_lng ,k_t_air ,my_lng_out,my_air_out,cp_lng_out,cp_air_out ,rho_air_out ,rho_lng_out,Re_lng,Pr_lng , Re_air ,Pr_air ,v_air ,v_lng]=PROPS_sp1(T_lng,P_lng ,T_air ,P_air ,mdot_lng,mdot_air, r_i ,r_o ,r_O, No_tubes,No_ribs,thickness_rib ,length_rib ,Ribs) ; 54%Calculate heat transfer parameters 55 [U,h_i,h_o,A_i,A_o,~,~,~,~,T_w,h_fin ,T_fin ,length_fin]=overall_heatcoeff_sp(Re_lng,Pr_lng,Re_air , Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,k_t_lng ,k_t_air ,No_tubes,Ribs,No_ribs,thickness_rib , length_rib ,T_air ,T_lng,No_rings,P_air ,v_air); 56%Compute new temperatures 57 deltaT_air_sp=U No_tubes/(mdot_air cp_air_out) ( T_air T_lng ) ;%Temperature * * * − difference of Air in Kelvin 58 T_air=T_air deltaT_air_sp; %New temperature − of Air 59 deltaT_lng_sp=U No_tubes/(mdot_lng cp_lng_out) ( T_air T_lng ) ;%Temperature * * * − difference of LNG in Kelvin 60 T_lng=T_lng+deltaT_lng_sp; %New temperature of LNG 61%Pressure drop calculations 62 rho_air_sp1(i+1)=rho_air_out;%Fill the endvector with first value 63 rho_lng_sp1(i+1)=rho_lng_out;%Fill the endvector with first value 64 [P_lng, P_air , delta_p_air , delta_p_lng]=Newpressure(P_lng,P_air ,Re_lng,Re_air,mdot_lng,mdot_air,dx, r_o, r_i ,r_O,rho_lng_sp1 ,rho_air_sp1 ,No_tubes,Ribs ,No_ribs,thickness_rib ,length_rib , i) ; 65%Data for plotting 66 data_save_sp1%Saving variables 67 counter =1: i +1;%Lenght count of the exchanger 68 co=counter *dx ;%calculate the acutal length 69 70%Bend pressure drop calculations 71 if co(end) == 0.62 || co(end) == 1.24%Location where the bend is calculated 72%LNG 73 u_lng1=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng_liq(T_lng ,P_lng) ;%Flow speed 74 delta_P_bend1_lng = bend_pressuredrop (0.4187,rho_lng_liq(T_lng,P_lng),u_lng1,2* r_i ,’LNG’,’bpc’);% Calcualte bend pressure drop 75 delta_P_lng_sp1(i)=delta_P_lng_sp1(i)+delta_P_bend1_lng;%Fill the vector with value 76 P_lng=P_lng delta_P_bend1_lng/100000;%Calculate new pressure − 106 C.C ODE

77%a i r 78 D_h_air =4 (pi r_O^2 No_tubes r_o^2 pi)/(2 pi (r_O+No_tubes r_o ) ) ;%Flow speed * * − * * * * * 79 u_air1=mdot_air /(pi ( r_O^2 r _ i ^2 No_tubes))/rho_air(T_air ,P_air) ;%speed calculation ignoring * − * possible fins 80 delta_P_bend1_air = bend_pressuredrop (0.4187,rho_air(T_air,P_air),u_air1,D_h_air,’Air’,’bpc’);% Calcualte bend pressure drop 81 delta_P_air_sp1(i)=delta_P_air_sp1(i)+delta_P_bend1_air;%Fill the vector with value 82 P_air=P_air delta_P_bend1_air/100000;%Calculate new pressure − 83 end 84 85 if sat_temp(P_lng)<= T_lng%break criterion saturation temperature 86 break 87 end 88 end 89 90%% Filling last or first positions of data 91 92 Walltemp_sp1(1) = Walltemp_sp1(2); 93 U_sp1(length (U_sp1) +1)=U_sp1(end); 94 h_i_sp1 (length(h_i_sp1)+1)=h_i_sp1(end); 95 h_o_sp1 (length(h_o_sp1)+1)=h_o_sp1(end); 96 delta_P_air_sp1 (length(delta_P_air_sp1)+1)=delta_P_air_sp1(end 1) ; − 97 delta_P_lng_sp1 (length(delta_P_lng_sp1)+1)=delta_P_lng_sp1(end 1) ; − 98 Reair_sp1 (length(Reair_sp1)+1)=Reair_sp1(length(Reair_sp1)); 99 Relng_sp1 (length(Relng_sp1)+1)=Relng_sp1(length(Relng_sp1)) ; 100 Prair_sp1 (length(Prair_sp1)+1)=Prair_sp1(length(Prair_sp1)); 101 Prlng_sp1 (length(Prlng_sp1)+1)=Prlng_sp1(length(Prlng_sp1)); 102 visc_lng_sp1(1)=my_lng(120,60); 103 rho_lng_sp1(1)=rho_lng(120,60); 104 quality ( 1 :length(co))=0; 105 Fin_T_profile_sp1(:,1)=Fin_T_profile_sp1(:,2);%erasing first line for better plots 106 v_air_sp1(1)=v_air_sp1(2); 107 v_lng_sp1(1)=v_lng_sp1(2); 108 109 110 elseif type ==’Counter flow’%If flow arrangement is counter flow − 111 for i=1:100000%Large value to create long loop (=while) 112%Determine fluid properties 113 [k_t_lng ,k_t_air ,my_lng_out,my_air_out,cp_lng_out,cp_air_out,rho_air_out ,rho_lng_out,Re_lng,Pr_lng ,Re_air ,Pr_air ,v_air ,v_lng]=PROPS_sp1(T_lng,P_lng,T_air ,P_air ,mdot_lng,mdot_air, r_i ,r_o ,r_O, No_tubes,No_ribs,thickness_rib ,length_rib ,Ribs) ; 114%Calculate heat transfer parameters 115 [U,h_i,h_o,A_i,A_o,~,~,~,~,T_w,h_fin,T_fin,length_fin]=overall_heatcoeff_sp(Re_lng,Pr_lng,Re_air, Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,k_t_lng ,k_t_air ,No_tubes,Ribs,No_ribs,thickness_rib , length_rib ,T_air ,T_lng,No_rings,P_air ,v_air); 116%Compute new temperatures 117 deltaT_air_sp=U No_tubes/(mdot_air cp_air_out) ( T_air T_lng ) ;%Temperature * * * − difference of Air in Kelvin 118 T_air=T_air+deltaT_air_sp;%New temperature of Air 119 deltaT_lng_sp=U No_tubes/(mdot_lng cp_lng_out) ( T_air T_lng ) ;%Temperature * * * − difference of LNG in Kelvin 120 T_lng=T_lng+deltaT_lng_sp;%New temperature of LNG 121%Pressure drop calculations 122 rho_air_sp1(i+1)=rho_air_out;%Fill the endvector with first value 123 rho_lng_sp1(i+1)=rho_lng_out;%Fill the endvector with first value 124 [P_lng, P_air, delta_p_air , delta_p_lng]=Newpressure(P_lng,P_air,Re_lng,Re_air,mdot_lng,mdot_air, dx,r_o,r_i ,r_O,rho_lng_sp1 ,rho_air_sp1 ,No_tubes,Ribs,No_ribs,thickness_rib ,length_rib , i) ; %Data for plotting 125 126%Data for plotting 127 data_save_sp1%Saving variables 128 129 counter =1: i +1;%Lenght count of the exchanger 130 co=counter *dx ;%calculate the acutal length 131 132%Bend pressure drop calculations 133 if co(end) == 0.62 || co(end) == 1.24%Location where the bend is calculated 134%LNG C.10. IMP_QUAN_PIPE 107

135 u_lng1=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng_liq(T_lng ,P_lng) ;%Flow speed 136 delta_P_bend1_lng = bend_pressuredrop (0.4187,rho_lng_liq(T_lng,P_lng),u_lng1,2* r_i ,’LNG’,’bpc ’);%Calcualte bend pressure drop 137 delta_P_lng_sp1(i)=delta_P_lng_sp1(i)+delta_P_bend1_lng;%Fill the vector with value 138 P_lng=P_lng delta_P_bend1_lng/100000;%Calculate new pressure − 139%Air 140 D_h_air =4 (pi r_O^2 No_tubes r_o^2 pi)/(2 pi (r_O+No_tubes r_o ) ) ;%Flow speed * * − * * * * * 141 u_air1=mdot_air /(pi ( r_O^2 r _ i ^2 No_tubes))/rho_air(T_air ,P_air) ;%speed calculation ignoring * − * possible fins 142 delta_P_bend1_air = bend_pressuredrop (0.4187,rho_air(T_air,P_air),u_air1,D_h_air,’Air’,’bpc’) ;%Calcualte bend pressure drop 143 delta_P_air_sp1(i)=delta_P_air_sp1(i)+delta_P_bend1_air;%Fill the vector with value 144 P_air=P_air delta_P_bend1_air/100000;%Calculate new pressure − 145 end 146 147 if sat_temp(P_lng)<= T_lng%break criterion saturation temperature 148 break 149 end 150 end 151 152 153%% Filling last positions of data 154 Walltemp_sp1(1) = Walltemp_sp1(2); 155 U_sp1(length (U_sp1) +1)=U_sp1(end); 156 h_i_sp1 (length(h_i_sp1)+1)=h_i_sp1(end); 157 h_o_sp1 (length(h_o_sp1)+1)=h_o_sp1(end); 158 delta_P_air_sp1 (length(delta_P_air_sp1)+1)=delta_P_air_sp1(end 1) ; − 159 delta_P_lng_sp1 (length(delta_P_lng_sp1)+1)=delta_P_lng_sp1(end 1) ; − 160 Reair_sp1 (length(Reair_sp1)+1)=Reair_sp1(length(Reair_sp1)); 161 Relng_sp1 (length(Relng_sp1)+1)=Relng_sp1(length(Relng_sp1)) ; 162 Prair_sp1 (length(Prair_sp1)+1)=Prair_sp1(length(Prair_sp1)); 163 Prlng_sp1 (length(Prlng_sp1)+1)=Prlng_sp1(length(Prlng_sp1)); 164 visc_lng_sp1(1)=my_lng(120,60); 165 rho_lng_sp1(1)=rho_lng(120,60); 166 quality ( 1 :length(co))=0; 167 Fin_T_profile_sp1(:,1)=Fin_T_profile_sp1(:,2);%erasing first line for better plots 168 v_air_sp1(1)=v_air_sp1(2); 169 v_lng_sp1(1)=v_lng_sp1(2); 170 171 end 172 173%% Add pressure drop due to entrance 174 delta_P_lng_sp1(1)=delta_P_lng_sp1(1)+delta_P_flatrate_lng1+delta_P_flatrate_lng2 ; 175 delta_P_air_sp1(1)=delta_P_air_sp1(1)+delta_P_flatrate_air ; 176 177%% Trigger Twophase flow 178 AccPressureloss

C.10. IMP_QUAN_PIPE

Program/imp_quan_pipe.m 1%Function to determine Reynolds and Prandtl numbers 2%INPUTS:rho, mdot,D_h,my,cp,k_t 3%OUTPUTS: Reynoldsnumber, Prandtlnumber 4 function[Re,Pr]=imp_quan_pipe(rho,v, r_i ,my,cp, k_t) 5 Re = rho*v*2* r _ i /my; 6 Pr=cp*my/ k_t ; 7 end

C.11. IMP_QUAN_SHELL

Program/imp_quan_shell.m 1%Function to determine Reynolds and Prantl number 2%INPUS:rho, mdot,D_h,my,cp,k_t 3%OUTPUTS: Reynoldsnumber, Prandtlnumber 108 C.C ODE

4 function[Re,Pr]=imp_quan_shell(rho,v,r_o ,r_O,my,cp,k_t ,No_tubes,No_ribs ,length_rib , thickness_rib ,Ribs) 5 if Ribs ==’n’ 6 P = ( r_O*2* pi+No_tubes *2* pi * r_o ) ;%wetted perimeter: circumference of Shell and outer circumference of the tubes inm 7 area = pi r_O^2 No_tubes r_o^2 pi; * − * * 8 elseif Ribs ==’y’ 9 P = ( r_O*2* pi+No_tubes *2* pi * r_o+No_ribs *No_tubes*2* length_rib);%wetted perimeter: circumference of Shell and outer circumference of the tubes inm 10 area = pi r_O^2 No_tubes r_o^2 pi No_tubes No_ribs thickness_rib length_rib; * − * * − * * * 11% Af=No_ribs *(2* length_rib+thickness_rib); 12% Ar=pi (r_o 2) No_ribs thickness_rib; * * − * 13% Ds=0.0269;%d_shell; 14% Sf=pi/4 (Ds^2 (r_o 2)^2) No_ribs length_rib thickness_rib; * − * − * * 15% d_h=4 * Sf/(Af+Ar) 16 end 17 d_h = 4* area/P ; 18 Re = rho*v*d_h/my; 19 Pr = cp*my/ k_t ; 20 end

C.12. LAYOUT_FINAL

Program/Layout_ﬁnal.m 1%% Layout program to detect overlaps and to show the current layout 2%The layout should be checked manually since the rounding option can lead 3%toa rounding error in display 4 5 figure 6 7%% Starting circles 8 cenmid=[0 0 ] ; 9 radmid=r_o; 10 censhell =[0 0 ] ; 11 radshell=r_O ; 12 13%% Bundle circles 14 if No_tubes > 1 15 radial_distance=(r_O)/(No_rings+0.5);%space between the rings 16 for i=1:No_rings 17 ring_circ ( i ) =2* pi * radial_distance * i ;%gives the circumferences of the rings 18 end 19 total_divisionlength=sum(ring_circ);%sums the circumferences of the rings 20 ring_arc=total_divisionlength/No_tubes;%gets the average spacing of the tubes 21 tubes_perring=floor(ring_circ/ring_arc);%retrieve the tubes per ring 22 ring_spacing =2* pi./tubes_perring;%project the tube distances ona ring of2pi 23 x=size(No_tubes,No_rings) ; 24 y=size(No_tubes,No_rings) ; 25 26 for u=1:No_rings 27 arc=ring_spacing(u); 28 r adi i (u)=u* radial_distance;%gives the radius of the actual ring 29 for i=1:tubes_perring(u)%loop until the maximum permitable tubes per ring are reached 30 x ( i , u)= r ad ii (u) * cos(arc);%gettingx coordiantes of the tube 31 y ( i , u)= r ad ii (u) * sin(arc);%gettingy coordinates of the tube 32 arc=arc+ring_spacing(u);%setting new arc 33 34 end 35 f =reshape(x,[],1);%sort allx in one column 36 f f =reshape(y,[],1);%sort ally in one column 37 P=[ f f f ] ; 38 39 end 40 radius =( r_o *ones (1 ,length(P)))’; 41 radii_rings =[1:u ] . * radial_distance; 42 centers_rings=zeros(u,2); 43 44%% Drawing circles 45%Shell and tube circles 46 v i s c i r c l e s (cenmid , radmid) C.13. MOMENTUM_PRESSURE_DROP 109

47%viscircles(cenmid,d_shell/2) 48 viscircles(censhell,radshell,’EdgeColor’,’g’) 49 grid off 50 axis equal 51%Bundle circles 52 viscircles(P,radius,’EdgeColor’,’r’) 53%Shellradius=ones(size(radius)) * d_shell/2 54%viscircles(P,Shellradius,’EdgeColor’,’b’,’LineStyle’,’:’) 55 56 xlabel(’Horizontal Length[m]’) 57 ylabel(’Vertical Length[m]’) 58 59%Draw the rings 60 viscircles(centers_rings,radii_rings,’LineStyle’,’ ’) −− 61 62 elseif No_tubes == 1 63 viscircles([0 0],r_o,’EdgeColor’,’r’) 64 viscircles([0 0],r_O,’EdgeColor’,’g’) 65 axis equal 66 xlabel(’Horizontal Length[m]’) 67 ylabel(’Vertical Length[m]’) 68 else disp(’Error: Unlogical number of tubes: Layout’) 69 return 70 end 71%% Ribs 72 if Ribs ==’y’ 73 if No_tubes > 1 74 rib_spacing =2* pi/No_ribs; 75 radius_rib_o=r_o+length_rib; 76 arc=rib_spacing ; 77 x = [ ] ; 78 y = [ ] ; 79 for u=1:No_ribs 80 x (u, 1 ) =r_o * cos(arc); 81 x(u,2)=radius_rib_o * cos(arc); 82 y (u, 1 ) =r_o * sin(arc); 83 y(u,2)=radius_rib_o * sin(arc); 84 arc=arc+rib_spacing; 85 end 86 line(x’,y’,’Color’,’b’) 87 for p=1:length(P) 88 xi= f (p)+x ; 89 yi= f f (p)+y ; 90 line(xi’,yi’,’Color’,’b’) 91 end 92 elseif No_tubes == 1 93 rib_spacing =2* pi/No_ribs; 94 radius_rib_o=r_o+length_rib; 95 arc=rib_spacing ; 96 x = [ ] ; 97 y = [ ] ; 98 for u=1:No_ribs 99 x (u, 1 ) =r_o * cos(arc); 100 x(u,2)=radius_rib_o * cos(arc); 101 y (u, 1 ) =r_o * sin(arc); 102 y(u,2)=radius_rib_o * sin(arc); 103 arc=arc+rib_spacing; 104 end 105 line(x’,y’,’Color’,’b’) 106 end 107 end

C.13. MOMENTUM_PRESSURE_DROP

Program/momentum_pressure_drop.m 1%momentum_pressure_drop calculates the momentum effect pressure drop ina round 2%pipe based on the density difference before and after 3 4%INPUTS: inlet density, outlet density 110 C.C ODE

5%OUPUTS: change in specific volume 6 7 function mom = momentum_pressure_drop(rho_in ,rho_out) 8 mom=1/rho_out 1/rho_in ; − 9 end

C.14. NEWPRESSURE

Program/Newpressure.m 1%function to determine the new pressure aftera unit section has been passed in single phase parts 2%INPUTS:P_lng,P_air,Re_lng,Re_air,mdot_lng,mdot_air,dx,r_o,r_i,r_O 3%OUTPUTS:P_lng, P_air 4 5 function [P_lng, P_air , delta_p_air_fric ,delta_p_lng_fric]=Newpressure(P_lng,P_air ,Re_lng,Re_air ,mdot_lng, mdot_air,dx,r_o, r_i ,r_O,density_lng ,density_air ,No_tubes,Ribs ,No_ribs,thickness_rib ,length_rib , Counter ) 6 7%% Define geometries 8 D_h_lng=2* r _ i ;%hydraulic diameter for tube 9 area_lng=pi * r _ i ^2;%free flow area of the tube 10 if Ribs ==’y’%Condition if fins area applied 11 area_air =(pi r_O^2) (No_tubes pi r_o ^2) No_tubes No_ribs length_rib thickness_rib; * − * * − * * * 12 else 13 area_air =(pi r_O^2) (No_tubes pi r_o ^2) ; * − * * 14 end 15 16%% Calculate pressure drop inside the LNG tubes 17%frictional pressure drop 18 delta_p_lng_fric=singlephase_pressure_pipe_lng(Re_lng ,D_h_lng,mdot_lng,area_lng ,dx,density_lng(Counter) , No_tubes ) ; 19%momentum pressure drop 20 if Counter>1%ensure no programm instabilities > ignorance of the effect the first iteration − 21 G_lng=mdot_lng/No_tubes/area_lng; 22 mom_lng = momentum_pressure_drop(density_lng(Counter 1),density_lng(Counter)) ; − 23 delta_p_lng_mom=G_lng^2*mom_lng; 24 else 25 delta_p_lng_mom=0; 26 end 27 28%% Calculate pressure drop outside the tubes (=in the shell) 29%frictional pressure drop 30 if No_tubes == 1%model if onlya double pipe exchanger is used 31 delta_p_air_fric=singlephase_pressure_pipe(Re_air ,mdot_air,area_air ,dx,density_air(Counter) ,Ribs, No_ribs,thickness_rib ,length_rib ,r_O,r_o ,No_tubes) ; 32 elseif No_tubes > 1%model if more than1 tube is used 33 delta_p_air_fric=singlephase_pressure_shell(Re_air ,mdot_air,area_air ,dx,density_air(Counter) ,No_tubes, r_o,r_O,Ribs,No_ribs,thickness_rib ,length_rib) ;%pressure drop only valid for annulus! extended type must be remodeled 34 else 35 disp(’Illegal number of tubes: Newpressure’) 36 return 37 end 38%momentum pressure drop 39 if Counter>1%ensure no programm instabilities > ignorance of the effect the first iteration − 40 G_air=mdot_air/area_air; 41 mom_air = momentum_pressure_drop(density_air(Counter 1),density_air(Counter)) ; − 42 delta_p_air_mom=G_air^2*mom_air ; 43 else 44 delta_p_air_mom=0; 45 end 46%% Calculate new pressure 47 P_lng=P_lng delta_p_lng_fric/100000 delta_p_lng_mom/100000;% new pressure of LNG − − 48 P_air=P_air delta_p_air_fric/100000 delta_p_air_mom/100000;% new pressure ofAIR − −

C.15. NEWPRESSURE_AIR

Program/Newpressure_air.m 1%function to determine the new pressure aftera unit section has been passed in single phase parts C.16. OVERALL_HEATCOEFF_SP 111

2%INPUTS:P_lng,P_air,Re_lng,Re_air,mdot_lng,mdot_air,dx,r_o,r_i,r_O 3%OUTPUTS:P_lng, P_air 4 5 function [P_air ,delta_p_air]=Newpressure_air(P_air ,Re_air ,mdot_air,dx,r_o,r_O,density_air ,No_tubes,Ribs , No_ribs ,thickness_rib ,length_rib ,Counter) 6 if Ribs ==’y’%Condition of fins are applied 7 area_air=(pi r_O^2) (No_tubes pi r_o ^2) No_tubes No_ribs length_rib thickness_rib; * − * * − * * * 8 elseif Ribs ==’n’ 9 area_air=(pi r_O^2) (No_tubes pi r_o ^2) ; * − * * 10 end 11 12%% Which model is applied? Annulus flow or tube bank flow? 13%fricitonal pressure drop 14 if No_tubes == 1 15 delta_p_air=singlephase_pressure_pipe(Re_air ,mdot_air,area_air ,dx,density_air(Counter) ,Ribs,No_ribs, thickness_rib ,length_rib ,r_O,r_o ,No_tubes) ;%Model for pipe also used for anulus 16 elseif No_tubes > 1 17 delta_p_air=singlephase_pressure_shell(Re_air ,mdot_air,area_air ,dx,density_air(Counter) ,No_tubes,r_o, r_O,Ribs,No_ribs,thickness_rib ,length_rib) ;%pressure drop only valid for annulus! extended type must be remodeled 18 else 19 disp(’Illegal number of tubes: Newpressure’) 20 return 21 end 22 23%momentum pressure drop 24 if Counter>2 25 G_air=mdot_air/area_air; 26 mom_air = momentum_pressure_drop(density_air(Counter 1),density_air(Counter)) ; − 27 delta_p_air_mom=G_air^2*mom_air ; 28 else 29 delta_p_air_mom=0; 30 end 31 32%% Calculate new pressure 33 P_air=P_air delta_p_air/100000 delta_p_air_mom/100000;%new pressure of air − −

C.16. OVERALL_HEATCOEFF_SP

Program/overall_heatcoeff_sp.m 1%Function to calculate the overall heat transfer coefficient ofa unit 2%section dx 3%INPUTS: d_tube,r_O,r_o,r_i,k_t_lng,k_t_tube,k_t_air,thickness,dx,rho_lng,rho_air,mdot_lng,mdot_air,my_lng ,my_air,cp_lng,cp_air 4%OUTPUTS:overall heat transfer coefficient, outer heat transfer 5%coefficient, inner sectional area, outer sectional area, LNG Reynolds 6%number, Air Reynolds number, LNG Prandtl number, Air Prandtl number, Wall 7%temperature, fin heat transfer coefficient, Temperature distribution fin, 8%length fin 9 function [U,h_i ,h_o,A_i,A_o,Re_lng,Re_air ,Pr_lng ,Pr_air ,T_w,h_fin ,T_fin ,length_fin]= overall_heatcoeff_sp( Re_lng,Pr_lng ,Re_air ,Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,k_t_lng ,k_t_air ,No_tubes,Ribs,No_ribs, thickness_rib ,length_rib ,T_air ,T_lng,No_rings,P_air ,v_air) 10%% Definition of the Hydraulic diamteter 11 d_tube=2* r _ i ; 12 if No_tubes>1 13 r_art_free_shell=(r_O (No_rings 2+1) r_o)/(No_rings 2+1)+ r _ i ;%artificial hydraulic diameter per unit − * * * a r t i f i c a l shell; calcualtion from the amount of tubes independent 14 d_h_art_free_shell=2 r_art_free_shell;% taken from http://www. engineeringtoolbox.com/hydraulic * − equivalent diameter d_458.html − − 15 d_shell=d_h_art_free_shell;%shell diameter is artificial arounda tube 16 if d_shell <= 0 17 s t r = [’Value of artificial diameter negativ:’ num2str(d_shell)’overall_heatcoeff’]; 18 disp(str) 19 return 20 end 21 elseif No_tubes == 1 22 d_shell =2*r_O ; 23 end 24 112 C.C ODE

25%% Calculation of heat transfer coefficients of primary heat transfer surfaces 26 h_i=singlephase_heat_tube(Re_lng,Pr_lng ,d_tube,k_t_lng) ; 27 h_o=singlephase_heat_shell(Re_air ,Pr_air ,d_shell ,k_t_air ,No_ribs,thickness_rib ,length_rib ,No_tubes,Ribs, r_o,r_O,dx,T_air ,T_lng,k_t_tube) ; 28 A_i=dx*2* pi * r _ i ; 29 A_o=dx*2* pi * r_o ; 30 31%% Calculation of heat transfer coefficients of secondary heat transfer surfaces 32 [q_fin ,T_w,T_fin ,length_fin ,n_f,h_fin] = rib_sp(dx,length_rib ,No_ribs,T_air ,T_lng,thickness_rib ,Re_air , Pr_air ,k_t_tube ,h_i ,h_o,r_i ,r_o,Pr_lng ,No_rings,No_tubes,r_O,P_air ,v_air); 33 A_fin =( No_ribs *(2* length_rib) *dx ) ;%tip ignored, since also head conduction area ignored in tube 34 if Ribs ==’y’ 35 U=(1/(1/(2* pi * r _ i *dx* h_i )+log(r_o/r_i)/(2 * pi * k_t_tube *dx ) +1/(2* pi * r_o *dx*h_o+A_fin * h_fin ) ) ) ;% Actually this isU *A fora single tube, http://www4.ncsu.edu/~doster/NE400/Text/HeatExchangers/ HeatExchangers.PDF 36 else h_fin=0; 37 U=(1/(1/(2* pi * r _ i *dx* h_i )+log(r_o/r_i)/(2 * pi * k_t_tube *dx ) +1/(2* pi * r_o *dx*h_o) ) ) ;%Actually this isU * A fora single tube, http://www4.ncsu.edu/~doster/NE400/Text/HeatExchangers/HeatExchangers.PDF 38 end

C.17. OVERALL_HEATCOEFF_TP

Program/overall_heatcoeff_tp.m 1%Function to calculate the overall heat transfer coefficient ofa unit 2%section dx 3%INPUTS: d_tube,r_O,r_o,r_i,k_t_lng,k_t_tube,k_t_air,thickness,dx,rho_lng,rho_air,mdot_lng,mdot_air,my_lng ,my_air,cp_lng,cp_air 4%OUTPUTS:overall heat transfer coefficient, inner heat transfer 5%coefficient, outer heat transfer coefficient, inner sectional area, 6%outer sectional area, Air Reynolds number, Air Prandtl number, heat flux, 7%Wall temperature, fin heat transfer coefficient, 8%Temperature distribution fin, length fin, flow speed LNG 9 10 function [U,h_i ,h_o,A_i,A_o,Re_air ,Pr_air ,q,T_w,h_fin ,T_fin ,length_fin ,v_lng]= overall_heatcoeff_tp(Re_air ,Pr_air ,r_O,r_o, r_i ,k_t_tube ,thickness ,dx,No_tubes,x_dot,Pc,P_air ,T_air ,P_lng,T_lng,mdot_lng,Counter, M,Ribs ,No_ribs,thickness_rib ,length_rib ,No_rings,Pr_lng ,T_w, v_air) 11%% Definition of the Hydraulic diamteter 12 if No_tubes>1 13 r_art_free_shell=(r_O (No_rings 2+1) r_o)/(No_rings 2+1)+ r _ i ;%artificial hydraulic diameter per unit − * * * a r t i f i c a l shell; calcualtion from the amount of tubes independent 14 d_h_art_free_shell=2 r_art_free_shell;% taken from http://www. engineeringtoolbox.com/hydraulic * − equivalent diameter d_458.html − − 15 d_shell=d_h_art_free_shell;%shell diameter is artificial arounda tube 16 if d_shell <= 0 17 s t r = [’Value of artificial diameter negativ:’ num2str(d_shell)’overall_heatcoeff’]; 18 disp(str) 19 return 20 end 21%d_shell=2 *r_O; 22 elseif No_tubes == 1 23 d_shell =2*r_O ; 24 end 25%% Calculation of heat transfer coefficients of primary heat transfer surfaces 26 27%Calculation of LNG heat transfer coefficient: 28 [h_i ,q,v_lng]=tp_heat(x_dot ,T_w,Pc,P_lng ,T_lng, r_i ,mdot_lng,No_tubes,Counter,M,dx) ;%heat transfer coefficient and flux per tube 29 30%Calculation of air heat transfer coefficent: 31 h_o=singlephase_heat_shell(Re_air ,Pr_air ,d_shell , kt_air(T_air ,P_air) ,No_ribs,thickness_rib ,length_rib , No_tubes,Ribs ,r_o ,r_O,dx,T_air ,T_lng,k_t_tube) ;%heat transfer coefficient base on reynoldsnumber 32 33 A_i=dx*2* pi * r _ i ; 34 A_o=dx*2* pi * r_o ; 35 36%% Calculation of heat transfer coefficients of secondary heat transfer surfaces 37 [q_fin ,T_w,T_fin ,length_fin ,n_f,h_fin] = rib_tp(dx,length_rib ,No_ribs,T_air ,T_lng,thickness_rib ,Re_air , Pr_air ,k_t_tube ,h_i ,h_o,r_i ,r_o,Pr_lng ,No_rings,No_tubes,r_O,P_air ,v_air); 38 A_fin =( No_ribs *(2* length_rib) *dx ) ;%tip ignored, since also head conduction area ignored in tube C.18.PROPS_ SP1 113

39 if Ribs ==’y’ 40 U=(1/(1/(2* pi * r _ i *dx* h_i )+log(r_o/r_i)/(2 * pi * k_t_tube *dx ) +1/(2* pi * r_o *dx*h_o+A_fin * h_fin ) ) ) ;% Actually this isU *A fora single tube, http://www4.ncsu.edu/~doster/NE400/Text/HeatExchangers/ HeatExchangers.PDF 41 else h_fin=0; 42 U=(1/(1/(2* pi * r _ i *dx* h_i )+log(r_o/r_i)/(2 * pi * k_t_tube *dx ) +1/(2* pi * r_o *dx*h_o) ) ) ;%Actually this isU * A fora single tube, http://www4.ncsu.edu/~doster/NE400/Text/HeatExchangers/HeatExchangers.PDF 43 end

C.18. PROPS_SP1

Program/PROPS_sp1.m 1%General property function, that determines the important characteristic 2%thermodynamic properties. Every Unit in standard form except mass, which 3%is kg and pressure which is usually bar unless otherwise indicated 4%INPUTS:T_lng, P_lng,T_air,P_air 5%OUTPUTS:k_t_lng,k_t_air,my_lng,my_air,cp_lng,cp_air,rho_air_out,rho_lng_out,Re_lng,Pr_lng,Re_air,Pr_air 6 7 function [k_t_lng , k_t_air ,my_lng_out,my_air_out,cp_lng_out ,cp_air_out ,rho_air_out ,rho_lng_out ,Re_lng, Pr_lng ,Re_air ,Pr_air ,v_air ,v_lng]=PROPS_sp1(T_lng, P_lng,T_air ,P_air ,mdot_lng,mdot_air, r_i ,r_o,r_O, No_tubes,No_ribs ,thickness_rib ,length_rib ,Ribs) 8 k_t_lng=kt_lng_liq(T_lng,P_lng);%k_t inW/(m.K) 9 k_t_air=kt_air(T_air,P_air);%k_t inW/(m.K) 10 my_lng_out=my_lng_liq(T_lng ,P_lng) ;%my in Pa.s 11 my_air_out=my_air(T_air ,P_air) ;%my in Pa.s 12 cp_lng_out=cp_lng_liq(T_lng,P_lng) ;%cp inJ/kgK 13 cp_air_out=cp_air(T_air ,P_air);%cp inJ/kgK 14 rho_air_out=rho_air(T_air ,P_air);%rho in kg/m3 15 rho_lng_out=rho_lng_liq(T_lng,P_lng) ;%rho in kg/m3 16 v_lng=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng_out ;%flow velocity inm/s withina single tube 17 v_air=mdot_air / ( (pi ( r_O^2 No_tubes r_o^2))+(No_tubes No_ribs length_rib thickness_rib))/rho_air_out; * − * * * * %flow velocity in the Shell 18 [Re_lng,Pr_lng]=imp_quan_pipe(rho_lng_out ,v_lng , r_i ,my_lng_out,cp_lng_out ,k_t_lng) ; 19 [Re_air , Pr_air]=imp_quan_shell(rho_air_out ,v_air ,r_o,r_O,my_air_out,cp_air_out , k_t_air ,No_tubes,No_ribs, length_rib ,thickness_rib ,Ribs);

C.19. PROPS_SP2

Program/PROPS_sp2.m 1%General property function, that determines the important characteristic 2%thermodynamic properties. Every Unit in standard form except mass, which 3%is kg and pressure which is usually bar unless otherwise indicated 4%INPUTS:T_lng, P_lng,T_air,P_air 5%OUTPUTS:k_t_lng,k_t_air,my_lng,my_air,cp_lng,cp_air,rho_air_out,rho_lng_out,Re_lng,Pr_lng,Re_air,Pr_air 6 7 function [k_t_lng , k_t_air ,my_lng_out,my_air_out,cp_lng_out ,cp_air_out ,rho_air_out ,rho_lng_out ,Re_lng, Pr_lng ,Re_air ,Pr_air ,v_air ,v_lng]=PROPS_sp2(T_lng, P_lng,T_air ,P_air ,mdot_lng,mdot_air, r_i ,r_o,r_O, No_tubes,No_ribs ,thickness_rib ,length_rib ,Ribs) 8 k_t_lng=kt_lng_gas(T_lng,P_lng);%k_t inW/(m.K) 9 k_t_air=kt_air(T_air,P_air);%k_t inW/(m.K) 10 my_lng_out=my_lng_gas(T_lng ,P_lng) ;%my in Pa.s 11 my_air_out=my_air(T_air ,P_air) ;%my in Pa.s 12 cp_lng_out=cp_lng_gas(T_lng,P_lng) ;%cp inJ/kgK 13 cp_air_out=cp_air(T_air ,P_air);%cp inJ/kgK 14 rho_air_out=rho_air(T_air ,P_air);%rho in kg/m3 15 rho_lng_out=rho_lng_gas(T_lng,P_lng) ;%rho in kg/m3 16 v_lng=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng_out ;%flow velocity inm/s withina single tube 17 v_air=mdot_air / ( (pi ( r_O^2 No_tubes r_o^2))+(No_tubes No_ribs length_rib thickness_rib))/rho_air_out; * − * * * * %flow velocity in the Shell 18 [Re_lng,Pr_lng]=imp_quan_pipe(rho_lng_out ,v_lng , r_i ,my_lng_out,cp_lng_out ,k_t_lng) ; 19 [Re_air , Pr_air]=imp_quan_shell(rho_air_out ,v_air ,r_o,r_O,my_air_out,cp_air_out , k_t_air ,No_tubes,No_ribs, length_rib ,thickness_rib ,Ribs);

C.20. RUNFILE

Program/Runﬁle.m 114 C.C ODE

1 clc 2 close all 3 clear 4%% INPUTS Heat exchanger − 5 6 r_i=0.013;%radius of tube inm 7 r_O=0.13;%radius of Shell inm 8 thickness=0.001;%Thickness of the tubes 9 r_o=r_i+thickness;%Outer radius of tube 10 No_hex_units =3;%Total number of heat exchanger units used 11 mdot_lng=0.96/No_hex_units;%mass flow of LNG in kg/s 12%mdot_air=4.07/No_hex_units;%mass flow of air in kg/s,from cooling module 13 k_t_tube =26;%th. conductivity of tube;(stainless steel 26) 14 P_lng_in =52;%Pressure in bar Lng stream 15 P_air_in =43.12392;%Air exit pressure HPC(from AHEAD) 16 T_air_in =890;%Air exit temperature HPC(from AHEAD) 17 T_lng_in =120;%LNG storage temperature 18%T_w=600;%initial wall temperature assumption 19 L=1;%length inm for discretization 20 Steps_sp =100;%Discretization steps 21 dx=L/Steps_sp ;%Step size 22 No_tubes=20;%Total number of tubes(watch roundings!) 23 No_rings =2;%Number of tube rings 24 Ribs=’y’;%Ribs used yes(y) or no(n) 25 No_ribs =16;%Number of fins per tube 26 thickness_rib=0.0009;%Fin thickness inm 27 length_rib =0.007;%Length of rib 28 r_entrance =0.05;%Hex entrance flange radius 29 Kc_air =0.39;%entrance loss of air, graphically determined 30 Ke_air =0.65;%e x i t loss of air,graphically determined 31 Kc_lng =0.45;%entrance loss of LNG, graphically determined 32 Ke_lng =0.65;%e x i t loss of LNG, graphically determined 33 L_free =0.3187;%straight flow length aftera bend 34 type=’Parallelflow’;%Arrangement:’Parallelflow’ or’Counter flow’ − 35 T_air_ass =600;%Assumed final air ptemperature 36 T_req=600; 37 Layout_final%Layout check 38 39%% INPUTS Engine simulation − 40 T4=1450;%TIT in Kelvin 41 no_blades_ngv=80;%Number of Vanes in NGV1 42 no_blades_hpt=144;%Number of Blades in HPT1 43 44%% Setting cooling parameters 45 T_max_inc=1366.15;%maximum allowable temperature for the Nickel allow of the blades(assumed: ngv and hpt have same material) 46 T_hex_output=T_air_ass ;%Required outlet temperature of heat exchanger for cooling 47 P_hex_output=P_air_in *0.9991;%Assumed output pressure of hex 48 P4=P_air_in *0 . 9 6 ;%Pressure in bar 49 50%% Cooling requirement calculations 51 52 [mdot1, mdot_split1, P_split1 , T_split1 , Q_in1, deltaQ1, converged1] = cooling( T4, P4*100000 , T_hex_output , P_hex_output*100000, 1, 1, T_max_inc );%run cooling program to determine the mass flow and temperature needed to cool the blade to the necessary degree(T4, P4, T_air, P_air, stageNr, statorRotorNr, T_max_inc) 53 mdot_cool_NGV1_sb=sum(mdot_split1) ;%calculating complete mass flow needed for NGV1 cooling 54 mdot_cool_NGV1=mdot_cool_NGV1_sb*no_blades_ngv 55 if isnan(Q_in1)==1 56 disp(’Imaginary Heat Flux: Cooling!’) 57 break 58 end 59 60 [mdot2, mdot_split2, P_split2 , T_split2 , Q_in2, deltaQ2, converged2] = cooling( T4, P4*100000 , T_hex_output , P_hex_output*100000, 1, 1, T_max_inc );%run cooling program to determine the mass flow and temperature needed to cool the blade to the necessary degree 61 mdot_cool_HPT1_sb=sum(mdot_split2) ;%calculating complete mass flow needed for HPT1 cooling 62 mdot_cool_HPT1=mdot_cool_HPT1_sb* no_blades_hpt 63 if isnan(Q_in2)==1 64 disp(’Imaginary Heat Flux: Cooling!’) 65 break C.20.R UNFILE 115

66 end 67 68 [mdot3, mdot_split3, P_split3 , T_split3 , Q_in3, deltaQ3, converged3] = cooling( T4, P4*100000 , T_hex_output , P_hex_output*100000, 1, 1, T_max_inc );%run cooling program to determine the mass flow and temperature needed to cool the blade to the necessary degree(T4, P4, T_air, P_air, stageNr, statorRotorNr, T_max_inc) 69 mdot_cool_NGV2_sb=sum(mdot_split3) ;%calculating complete mass flow needed for NGV2 cooling 70 mdot_cool_NGV2=mdot_cool_NGV2_sb*no_blades_ngv ; 71 if isnan(Q_in3)==1 72 disp(’Imaginary Heat Flux: Cooling!’) 73 break 74 end 75 76 [mdot4, mdot_split4, P_split4 , T_split4 , Q_in4, deltaQ4, converged4] = cooling( T4, P4*100000 , T_hex_output , P_hex_output*100000, 1, 1, T_max_inc );%run cooling program to determine the mass flow and temperature needed to cool the blade to the necessary degree 77 mdot_cool_HPT2_sb=sum(mdot_split4) ;%calculating complete mass flow needed for HPT2 cooling 78 mdot_cool_HPT2=mdot_cool_HPT2_sb* no_blades_hpt ; 79 if isnan(Q_in4)==1 80 disp(’Imaginary Heat Flux: Cooling!’) 81 break 82 end 83 84%% Saving the required mass flow for the heat exchanger 85 clearing%Clear variables from last iteration 86 mdot_air=(mdot_cool_NGV1+mdot_cool_HPT1+mdot_cool_NGV2+mdot_cool_HPT2) /No_hex_units ; 87 hex_run%run heat exchanger module 88 P_hex_real=P_air ;%Set output pressure of hex to be real pressure 89 90%% Getting Averages 91%Stage1 92 T_cool_NGV1=sum(mdot_split1. * T_split1 )/mdot_cool_NGV1_sb;%calculating average temperature of cooling air 93 P_cool_NGV1=sum(mdot_split1. * P_split1 )/mdot_cool_NGV1_sb;%calculating average pressure of cooling air, again in Pa 94 T_cool_HPT1=sum(mdot_split2. * T_split2 )/mdot_cool_HPT1_sb;%calculating average temperature of cooling air 95 P_cool_HPT1=sum(mdot_split2. * P_split2 )/mdot_cool_HPT1_sb;%calculating average pressure of cooling air, in Pa 96 97%Stage2 98 T_cool_NGV2=sum(mdot_split3. * T_split3 )/mdot_cool_HPT2_sb;%calculating average temperature of cooling air 99 P_cool_NGV2=sum(mdot_split3. * P_split3 )/mdot_cool_HPT2_sb;%calculating average pressure of cooling air, in Pa 100 T_cool_HPT2=sum(mdot_split4. * T_split4 )/mdot_cool_HPT2_sb;%calculating average temperature of cooling air 101 P_cool_HPT2=sum(mdot_split4. * P_split4 )/mdot_cool_HPT2_sb;%calculating average pressure of cooling air, in Pa 102 103 Error_Pressure_assumption=P_hex_output P_hex_real%Error in initial pressure assumption for − iteration 104 delta_P_hex=(P_air_in P_hex_real) 100000%Absolute error of initial pressure assumption − * 105 delta_P_cool=(P_hex_output 100000 (P_cool_NGV1 mdot_cool_NGV1_sb+P_cool_NGV2 mdot_cool_NGV2_sb+P_cool_HPT2 * − * * *mdot_cool_HPT2_sb+P_cool_HPT1*mdot_cool_HPT1_sb) /(mdot_cool_HPT1_sb+mdot_cool_HPT2_sb+ mdot_cool_NGV1_sb+mdot_cool_NGV2_sb) )%pressure drop due to cooling 106 Total_Pressuredrop=delta_P_hex+delta_P_cool%Pressure drop over the exchanger and turbine blade 107 Allowable_Pressuredrop=P_air_in *100000*0.04%Allowable pressure drop 108 109 Cc=sum(cp_lng_tot)/Total_Length(end)/dx *mdot_lng%Average Cc 110 Ch=sum(cp_air_tot)/Total_Length(end)/dx * mdot_air 111 if Cc < Ch 112 Cmin=Cc 113 elseif Ch < Cc 114 Cmin=Ch 115 else disp(’Error?’) 116 end 117 eta=Cc/Cmin ( T_lng T_lng_in)/(T_air_in T_lng_in ) * − − 118 119 totalPlot2 120 Pressureplot 116 C.C ODE

C.21. SAT_TEMP

Program/sat_temp.m 1% saturation temperature asa function of pressure 2%Pressures from 20 52 bar − 3%INPUTS: pressure in bar 4%OUTPUTS: temperature in kelvin 5 6 function boil = sat_temp(P) 7 P=P*100000; 8 fun=[167.98 2000000.00 9 169.34 2100000.00 10 170.67 2200000.00 11 171.95 2300000.00 12 173.19 2400000.00 13 174.40 2500000.00 14 175.58 2600000.00 15 176.73 2700000.00 16 177.85 2800000.00 17 178.94 2900000.00 18 180.01 3000000.00 19 181.06 3100000.00 20 182.09 3200000.00 21 183.09 3300000.00 22 184.07 3400000.00 23 185.04 3500000.00 24 185.99 3600000.00 25 186.93 3700000.00 26 187.85 3800000.00 27 188.75 3900000.00 28 189.65 4000000.00 29 190.53 4100000.00 30 191.40 4200000.00 31 192.26 4300000.00 32 193.12 4400000.00 33 193.96 4500000.00 34 194.81 4600000.00 35 195.65 4700000.00 36 196.50 4800000.00 37 197.35 4900000.00 38 198.22 5000000.00 39 199.12 5100000.00 40 200.06 5200000.00]; 41 42 boil=qinterp1(fun(:,2) ,fun(:,1) ,P);

C.22. SINGLEPHASE_HEAT_SHELL

Program/singlephase_heat_shell.m 1%Function to determine the single phase heat transfer for the(artificial)shell 2%INPUTS: Reynoldsnumber, Prandtlnumber hydraulic diameter and conductivity 3%OUTPUTS:heat transfer coefficient 4 function[h]=singlephase_heat_shell(Re_air ,Pr_air ,d_shell ,k_t ,No_ribs,thickness_rib ,length_rib ,No_tubes, Ribs ,r_o,r_O,dx,T_air ,T_lng,k_t_tube) 5 if Ribs ==’n’%standard anulus flow 6 if Re_air<=3000 7%Nu=4.364;%laminar Nu at constant heat flux 8 Nu=3.66;%laminar Nu at constant Wall temp 9 else 10 f =(0.79 log(Re_air) 1.64)^ 2;%Darcy friction factor * − − 11 Nu=(( f /8) ( Re_air 1000) Pr_air)/(1+12.7 ( f /8) ^0.5 (Pr_air^(2/3) 1) ) ;%turbulent Nu for smaller than5e6 * − * * * − 12 end 13 14%Complete Shell 15%A=r_O^2 pi No_tubes r_o^2 pi; * − * * 16%P=2 * pi *r_O+No_tubes *2* pi * r_o; 17 18%Artificial Shell C.23. SINGLEPHASE_HEAT_TUBE 117

19 A=( d_shell /2) ^2 pi r_o^2 pi; * − * 20 P=2* pi *(d_shell/2)+2* pi * r_o ; 21 D_h=4*A/P; 22 h=Nu* k_t /D_h; 23 24 elseif Ribs ==’y’%finned flow, implemented according to the paper from Jerry Taborek 25% Af=No_ribs *(2* length_rib+thickness_rib); 26% Ar=pi (r_o 2) No_ribs thickness_rib; * * − * 27% Ds=d_shell%2 *r_O; 28% Sf=pi/4 (Ds^2 (r_o 2)^2) No_ribs length_rib thickness_rib; * − * − * * 29% D_h=4 * Sf/(Af+Ar) 30% d_shell 31 32 33%Complete Shell 34%A=r_O^2 pi No_tubes r_o^2 pi No_tubes No_ribs length_rib thickness_rib; * − * * − * * * 35%P=2 * pi *r_O+No_tubes *2* pi * r_o+No_tubes * No_ribs *2* length_rib; 36 37%Only Artificial Shell with ribs in Dh 38%A=(d_shell/2)^2 pi r_o^2 pi No_ribs length_rib thickness_rib; * − * − * * 39%P=2 * pi *(d_shell/2)+2 * pi * r_o+No_ribs *2* length_rib; 40% D_h=4 *A/P; 41% Artificial Shell without ribs in Dh 42 A=( d_shell /2) ^2 pi r_o^2 pi; * − * 43 P=2* pi *(d_shell/2)+2* pi * r_o ; 44 D_h=4*A/P; 45 f =(0.79 log(Re_air) 1.64)^ 2;%Darcy friction factor * − − 46 Nu=(( f /8) ( Re_air 1000) Pr_air)/(1+12.7 ( f /8) ^0.5 (Pr_air^(2/3) 1) ) ;%turbulent Nu for smaller than5e6 * − * * * − 47 h=Nu* k_t /D_h; 48 49% Nu_t=((f/8) Re_air Pr_air)/(1.07+12.7 (f/8)^0.5 (Pr_air^(2/3) 1)); * * * * − 50% Nu_an=Nu_t *0.86*(d_shell/2)/r_o; 51%h=Nu_an * k_t/D_h; 52 end

C.23. SINGLEPHASE_HEAT_TUBE

Program/singlephase_heat_tube.m 1%Function to determine the single phase heat transfer 2%INPUTS: Reynoldsnumber, Prandtlnumber hydraulic diameter and conductivity 3%OUTPUTS:heat transfer coefficient 4 function[h]=singlephase_heat_tube(Re,Pr,D_h, k_t) 5 if Re<=3000 6%Nu=4.364;%laminar Nu at constant heat flux 7 Nu=3.66;%laminar Nu at constant Wall temp 8 else 9 f =(0.79 log(Re) 1.64)^ 2;%Darcy friction factor * − − 10 Nu=(( f /8) (Re 1000) Pr)/(1+12.7 ( f /8) ^0.5 ( Pr ^(2/3) 1) ) ;%turbulent Nu for smaller than5e6 * − * * * − 11 end 12 h=Nu* k_t /D_h; 13%St=Nu/Re/Pr;%Stanton number, maybe for later 14 end

C.24. SINGLEPHASE_PRESSURE_PIPE

Program/singlephase_pressure_pipe.m 1%Function to determine the single phase pressure drop in an annulus 2%Inputs: Reynoldsnumber, hydraulic diameter, mass flow, tube crossetional 3%area, length dx 4%Outputs:pressure drop 5 function[delta_p]=singlephase_pressure_pipe(Re,mdot,area ,dx,rho,Ribs ,No_ribs ,thickness_rib ,length_rib ,r_O, r_o ,No_tubes) 6 G=mdot/area ; 7 if Re<=2300 8 f =64/Re ;%laminar flow 9 elseif 2300

11 f2 =0.25 (log10((150.39/Re^0.98865) (152.66/Re)))^ 2;%turbulent flow * − − 12 Re_perc=(Re 2300)/7/100;%determine the percent wise share of the intermediate part − 13 f=f1 (1 Re_perc)+f2 Re_perc ;%interpolate weighted * − * 14 elseif 3000

C.25. SINGLEPHASE_PRESSURE_PIPE_LNG

Program/singlephase_pressure_pipe_lng.m 1%Function to determine the single phase pressure drop ina round pipe 2%Inputs: Reynoldsnumber, hydraulic diameter, mass flow, tube crossetional 3%area, length dx 4%Outputs:pressure drop 5 function[delta_p]=singlephase_pressure_pipe_lng(Re,D_h,mdot,area ,dx,rho,No_tubes) 6 G=mdot/No_tubes/area ; 7 if Re<=2300 8 f =64/Re ;%laminar flow 9 elseif 2300

C.26. SINGLEPHASE_PRESSURE_SHELL

Program/singlephase_pressure_shell.m 1%Function to determine the single phase pressure drop ina round pipe 2%Inputs: Reynoldsnumber, hydraulic diameter, mass flow, tube crossetional 3%area, length dx 4%Outputs:pressure drop 5 function[delta_p]=singlephase_pressure_shell(Re,mdot,area ,dx,rho_air_out ,No_tubes,r_o ,r_O,Ribs ,No_ribs , thickness_rib ,length_rib) 6 G=mdot/area ; 7 if Re<=2300 8 f =64/Re ;%laminar flow 9 elseif 2300

12 Re_perc=(Re 2300)/7/100;%determine the percent wise share of the intermediate part − 13 f=f1 (1 Re_perc)+f2 Re_perc ;%interpolate weighted * − * 14 elseif 3000

C.27. SPPHASETWO

Program/spphasetwo.m 1%% Second single phase flow subroutine 2 3 i =1;%initial iteration value 4 rho_air_sp2(1)=rho_air_tp(end);%Fill the endvector with first value 5 rho_lng_sp2(1)=rho_lng_tp(end);%Fill the endvector with first value 6 if type ==’Counter flow’%If flow arrangement is counter flow − 7 while T_air <= T_air_in%As long as the the input temperature is not reached 8%Determine fluid properties 9 [k_t_lng ,k_t_air ,my_lng_out,my_air_out,cp_lng_out,cp_air_out,rho_air_out ,rho_lng_out,Re_lng,Pr_lng ,Re_air ,Pr_air ,v_air ,v_lng]=PROPS_sp2(T_lng,P_lng,T_air ,P_air ,mdot_lng,mdot_air, r_i ,r_o ,r_O, No_tubes,No_ribs,thickness_rib ,length_rib ,Ribs) ; 10%Calculate heat transfer parameters 11 [U,h_i,h_o,A_i,A_o,~,~,~,~,T_w,h_fin,T_fin,length_fin]=overall_heatcoeff_sp(Re_lng,Pr_lng,Re_air, Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,k_t_lng ,k_t_air ,No_tubes,Ribs,No_ribs,thickness_rib , length_rib ,T_air ,T_lng,No_rings,P_air ,v_air); 12%Compute new temperatures 13 deltaT_air_sp2=U No_tubes/(mdot_air cp_air_out) ( T_air T_lng ) ;%Temperature * * * − difference of Air in Kelvin 14 T_air=T_air+deltaT_air_sp2;%New temperature of Air 15 deltaT_lng_sp2=U No_tubes/(mdot_lng cp_lng_out) ( T_air T_lng ) ;%Temperature * * * − difference of LNG in Kelvin 16 T_lng=T_lng+deltaT_lng_sp2;%New temperature of LNG 17%Pressure drop calculations 18 rho_air_sp2(i+1)=rho_air_out;%Fill the endvector with first value 19 rho_lng_sp2(i+1)=rho_lng_out;%Fill the endvector with first value 20 [P_lng, P_air, delta_p_air , delta_p_lng]=Newpressure(P_lng,P_air,Re_lng,Re_air,mdot_lng,mdot_air, dx,r_o,r_i ,r_O,rho_lng_sp2 ,rho_air_sp2 ,No_tubes,Ribs,No_ribs,thickness_rib ,length_rib , i) ; 21 22 23%Bend pressure drop calculations 24 if round((Total_Length(end)+i *dx ) *100)/100 == 1.99%last bend is prior to the actual one which is actually in the2pregion 25%LNG 26 u_lng1=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng_liq(T_lng ,P_lng) ;%Flow speed 27 delta_P_bend1_lng = bend_pressuredrop (L_free,rho_lng_liq(T_lng,P_lng),u_lng1,2* r_i ,’LNG’,’bpc ’);%Calcualte bend pressure drop 28 delta_P_lng_sp2(i)=delta_P_lng_sp2(i)+delta_P_bend1_lng;%Fill the vector with value 29 P_lng=P_lng delta_P_bend1_lng/100000;%Calculate new pressure − 30%Air 31 D_h_air =4 (pi r_O^2 No_tubes r_o^2 pi)/(2 pi (r_O+No_tubes r_o ) ) ;%Flow speed * * − * * * * * 120 C.C ODE

32 u_air1=mdot_air /(pi ( r_O^2 r _ i ^2 No_tubes))/rho_air(T_air ,P_air) ;%speed calculation ignoring * − * possible fins 33 delta_P_bend1_air = bend_pressuredrop (L_free,rho_air(T_air,P_air),u_air1,D_h_air,’Air’,’bpc’) ;%Calcualte bend pressure drop 34 delta_P_air_sp2(i)=delta_P_air_sp2(i)+delta_P_bend1_air;%Fill the vector with value 35 P_air=P_air delta_P_bend1_air/100000;%Calculate new pressure − 36 37 end 38%Data for plotting 39 data_save_sp2 40 41 i = i +1;%Set next iteration step 42 if T_air T_lng< 0.1%Breaking requirement in case Tlng=Tair − 43 break 44 end 45 end 46 counter2 =1:(length(T_air_sp2));%Lenght count of the exchanger 47 co2=counter2 *dx+Total_Length(end);%calculate the acutal length 48 49 50 elseif type ==’Parallelflow’ 51 while T_air >= T_req 52%Determine fluid properties 53 [k_t_lng ,k_t_air ,my_lng_out,my_air_out,cp_lng_out,cp_air_out,rho_air_out ,rho_lng_out,Re_lng,Pr_lng ,Re_air ,Pr_air ,v_air ,v_lng]=PROPS_sp2(T_lng,P_lng,T_air ,P_air ,mdot_lng,mdot_air, r_i ,r_o ,r_O, No_tubes,No_ribs,thickness_rib ,length_rib ,Ribs) ; 54%Calculate the overall heat transfer coefficient 55 [U,h_i,h_o,A_i,A_o,~,~,~,~,T_w,h_fin,T_fin,length_fin]=overall_heatcoeff_sp(Re_lng,Pr_lng,Re_air, Pr_air ,r_O,r_o,r_i ,k_t_tube ,thickness ,dx,k_t_lng ,k_t_air ,No_tubes,Ribs,No_ribs,thickness_rib , length_rib ,T_air ,T_lng,No_rings,P_air ,v_air); 56%Compute new temperatures 57 deltaT_air_sp2=U No_tubes/(mdot_air cp_air_out) ( T_air T_lng ) ;%Temperature * * * − difference of Air in Kelvin 58 T_air=T_air deltaT_air_sp2;%New temperature − of Air 59 deltaT_lng_sp2=U No_tubes/(mdot_lng cp_lng_out) ( T_air T_lng ) ;%Temperature * * * − difference of LNG in Kelvin 60 T_lng=T_lng+deltaT_lng_sp2;%New temperature of LNG 61%Pressure drop calculations 62 rho_air_sp2(i+1)=rho_air_out;%Fill the endvector with first value 63 rho_lng_sp2(i+1)=rho_lng_out;%Fill the endvector with first value 64 [P_lng, P_air, delta_p_air , delta_p_lng]=Newpressure(P_lng,P_air,Re_lng,Re_air,mdot_lng,mdot_air, dx,r_o,r_i ,r_O,rho_lng_sp2 ,rho_air_sp2 ,No_tubes,Ribs,No_ribs,thickness_rib ,length_rib , i) ; 65 66%Bend pressure drop calculations 67 if round((Total_Length(end)+i *dx ) *100)/100 == 1.99%last bend is prior to the actual one which is actually in the2pregion 68%LNG 69 u_lng1=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng_liq(T_lng ,P_lng) ;%Flow speed 70 delta_P_bend1_lng = bend_pressuredrop (L_free,rho_lng_liq(T_lng,P_lng),u_lng1,2* r_i ,’LNG’,’bpc ’);%Calcualte bend pressure drop 71 delta_P_lng_sp2(i)=delta_P_lng_sp2(i)+delta_P_bend1_lng;%Fill the vector with value 72 P_lng=P_lng delta_P_bend1_lng/100000;%Calculate new pressure − 73%Air 74 D_h_air =4 (pi r_O^2 No_tubes r_o^2 pi)/(2 pi (r_O+No_tubes r_o ) ) ;%Flow speed * * − * * * * * 75 u_air1=mdot_air /(pi ( r_O^2 r _ i ^2 No_tubes))/rho_air(T_air ,P_air) ;%speed calculation ignoring * − * possible fins 76 delta_P_bend1_air = bend_pressuredrop (L_free,rho_air(T_air,P_air),u_air1,D_h_air,’Air’,’bpc’) ;%Calcualte bend pressure drop 77 delta_P_air_sp2(i)=delta_P_air_sp2(i)+delta_P_bend1_air;%Fill the vector with value 78 P_air=P_air delta_P_bend1_air/100000;%Calculate new pressure − 79 80 end 81%Data for plotting 82 data_save_sp2 83 84 i = i +1;%Set next iteration step C.27. SPPHASETWO 121

85 if T_air T_lng< 0.1%Breaking requirement in case Tlng=Tair − 86 break 87 end 88 end 89 counter2 =1:(length(T_air_sp2));%Lenght count of the exchanger 90 co2=counter2 *dx+Total_Length(end);%calculate the acutal length 91 end 92 93%% Appending 94 95%% Calculations before enterring the exchanger 96%LNG e x i t into head − 97 v1_lng=mdot_lng /(pi * r _ i ^2*No_tubes)/rho_lng(T_lng ,P_lng) ;%initial flow speed 98 v2_lng=mdot_lng /(pi *r_O^2)/rho_lng(T_lng ,P_lng) ;%second flow speed 99 delta_P_flatrate_lng3 = flatratepressure(rho_lng(T_lng,P_lng) ,v1_lng,v2_lng,Ke_lng);%pressure drop 100 P_lng=P_lng delta_P_flatrate_lng3/100000;%set new pressure − 101 102%LNG e x i t into pipe − 103 v1_lng=mdot_lng /(pi *r_O^2)/rho_lng(T_lng ,P_lng) ;%initial flow speed 104 v2_lng=mdot_lng /(pi * r_entrance^2)/rho_lng(T_lng ,P_lng) ;%second flow speed 105 delta_P_flatrate_lng4 = flatratepressure(rho_lng(T_lng,P_lng) ,v1_lng,v2_lng,Ke_lng);%pressure drop 106 P_lng=P_lng delta_P_flatrate_lng4/100000;%set new pressure − 107 108%a i r 109 v2_air=mdot_air /(pi * r_entrance^2)/rho_air(T_air ,P_air) ;%initial flow speed 110 v1_air=mdot_air /(pi ( r_O^2 r _ i ^2 No_tubes))/rho_air(T_air ,P_air) ;%second flow speed * − * 111 delta_P_flatrate_air = flatratepressure(rho_air(T_air ,P_air) ,v1_air ,v2_air ,Ke_air);%pressure drop 112 P_air=P_air delta_P_flatrate_air/100000;%set new pressure − 113 114%% Filling last positions of data, clear data and append variables 115 116 P_lngdata_sp2(1)=P_lng_total(end); 117 P_lng_total=[P_lng_total P_lngdata_sp2]; 118 P_airdata_sp2(1)=P_air_total(end); 119 P_air_total = [P_air_total P_airdata_sp2]; 120 T_lng_sp2(1)=T_lng_total(end); 121 T_final_lng = [T_lng_total T_lng_sp2]; 122 T_air_sp2(1)=T_air_total(end); 123 T_final_air = [T_air_total T_air_sp2]; 124 U_sp2(end+1)=U_sp2(end); 125 U_final = [ U_total U_sp2 ] ; 126 h_i_sp2 (end+1)=h_i_sp2(end); 127 h_i_final = [h_i_total h_i_sp2]; 128 h_o_sp2 (end+1)=h_o_sp2(end); 129 h_o_final = [h_o_total h_o_sp2]; 130 X_dot_final = [X_dot_total x_dot]; 131 deltaTair_sp2(1) = deltaT_air_total(end); 132 deltaTlng_sp2(1) = deltaT_lng_total(end); 133 deltaT_air_total = [deltaT_air_total deltaTair_sp2]; 134 deltaT_lng_total = [deltaT_lng_total deltaTLNG_sp2]; 135 Reair_sp2 (end+1)=Reair_sp2(end); 136 Reynolds_air_final = [Reynolds_air_total Reair_sp2]; 137 Prair_sp2 (end+1)=Prair_sp2(end); 138 Prandtl_air_final = [Prandtl_air_total Prair_sp2]; 139 Total_Length = [Total_Length co2]; 140 Walltemp_sp2(1)=Walltemp_sptp(end); 141 Total_Walltemp = [Walltemp_sptp Walltemp_sp2]; 142 Fin_h_sp2(1) = Fin_h_tp(end); 143 Total_h_fin = [Total_h_fin Fin_h_sp2]; 144 Fin_T_profile_sp2(: ,1)=Fin_T_profile_sp2(: ,2) ; 145 Total_fin_t = [Total_fin_t Fin_T_profile_sp2]; 146 delta_P_lng_sp2(2)=delta_P_lng_sp2(3) ; 147 delta_P_lng_sp2(1)=delta_P_lng_tp(end); 148 delta_P_lng_sp2 (end)=delta_P_lng_sp2(end)+delta_P_flatrate_lng3+delta_P_flatrate_lng4 ; 149 delta_P_air_sp2(1)=delta_P_air_tp(end); 150 delta_P_air_sp2 (end)=delta_P_air_sp2(end)+delta_P_flatrate_air; 151 delta_P_tot_lng=[delta_P_tot_lng delta_P_lng_sp2]; 152 delta_P_tot_air=[delta_P_tot_air delta_P_air_sp2]; 153 v_air_sp2(1)=v_air_sp2(2); 154 v_lng_sp2(1)=v_lng_sp2(2) ; 155 v_lng_tot=[v_lng_tot v_lng_sp2]; 122 C.C ODE

156 v_air_tot=[v_air_tot v_air_sp2]; 157 rho_air_tot=[rho_air_tot rho_air_sp2]; 158 rho_lng_tot=[rho_lng_tot rho_lng_sp2]; 159 cp_air_tot=[cp_air_tot_sp1 cp_air_tot_tp cp_air_tot_sp2]; 160 cp_lng_tot=[cp_lng_tot_sp1 cp_lng_tot_tp cp_lng_tot_sp2]; 161 Relng_sp2(1)=Relng_sp2(2) ; 162 Prlng_sp2(1)=Prlng_sp2(2) ;

C.28. TOTALPLOT2

Program/totalPlot2.m 1%% Plots2 2 3 close all 4%%Heat Transfer 5 figure(’name’,’Heat Transfer1’); 6 subplot(1,2,1) 7 plot(Total_Length , T_final_air ,Total_Length , T_final_lng ,Total_Length ,Total_Walltemp) 8 grid on 9 xlabel(’Length[m]’) 10 ylabel(’Temperature[K]’) 11 legend(’T_{Air}’,’T_{LNG}’,’T_{Wall}’,’Location’,’NorthWest’) 12 title(’a) Temperature distribution’) 13 14 subplot(1,2,2) 15 plot(Total_Length ,U_final) 16 grid on 17 xlabel(’Length[m]’) 18 ylabel(’Total Heat Transfer[W/K]’) 19 title(’b) Heat transfer per step’) 20 21 figure(’name’,’Heat Transfer2’); 22 subplot(1,2,1) 23 semilogy(Total_Length , h_i_final) 24 grid on 25 xlabel(’Length[m]’) 26 ylabel(’Inner Heat Transfer Coefficient[W/(m^2K)]’) 27 title(’a) Heat transfer distribution tube’) 28 29 subplot(1,2,2) 30 plot(Total_Length ,h_o_final) 31 grid on 32 xlabel(’Length[m]’) 33 ylabel(’Outer Heat Transfer Coefficient[W/(m^2K)]’) 34 title(’b) Heat transfer distribution shell’) 35 36% subplot(1,3,3) 37% Total_h_fin(1)=Total_h_fin(2); 38% plot(Total_Length,Total_h_fin) 39% grid on 40% xlabel(’Length[m] ’ ) 41% ylabel(’Fin Heat Transfer Coefficient[W/(m2.K)]’) 42% title(’Heat transfer distribution fin’) 43 44%% Pressure 45% figure(’name’,’Pressure’); 46% ax= plotyy(Total_Length,delta_P_tot_air, Total_Length,delta_P_tot_lng); 47% grid on 48% xlabel(’Length[m] ’ ) 49% set(get(ax(2),’Ylabel’),’String’,’LNG Pressure drop in[Pa]’) 50% set(get(ax(1),’Ylabel’),’String’,’Air Pressure drop in[Pa]’) 51% legend(’Air pressure drop’,’LNG pressure drop’) 52% title(’Pressure drop distribution’) 53 54 55%% Phase Change 56 57 figure(’name’,’Phase Change1’); 58 subplot(1,3,1) C.28. TOTALPLOT2 123

59 plot(Total_Length ,X_dot_final) 60 grid on 61 xlabel(’Length[m]’) 62 ylabel(’Vapor Quality’) 63 title(’a) Vapor quality distribution’) 64 65 subplot(1,3,2) 66 plot(Total_Length ,[ quality alpha.lng x_dot]) 67 grid on 68 ylabel(’Void Fraction\alpha’) 69 xlabel(’Length[m]’) 70 title(’b) Void fraction distribution’) 71 72 subplot(1,3,3) 73 plot(vapor_quality.lng ,alpha.lng) 74 grid on 75 xlabel(’Void Fraction\alpha’) 76 ylabel(’Vapor quality’) 77 title(’c) Vapor quality over Void fraction’) 78% 79% delta_Pa.lng(1)=delta_Pa.lng(2); 80% figure(’name’,’Phase Change 2’); 81% subplot(1,2,1) 82% plot(temp,delta_Pa.lng+delta_Pf.lng) 83% grid on 84% xlabel(’Length[m] ’ ) 85% ylabel(’Total Pressure drop[Pa]’) 86% title(’a) Total two phase pressure drop’) 87% subplot(1,2,2) 88% semilogy(temp,delta_Pa.lng, temp,delta_Pf.lng) 89% grid on 90% xlabel(’Length[m] ’ ) 91% ylabel(’Pressure drop[Pa]’) 92% legend(’Accelerational’,’Frictional’,’Location’,’SouthEast’) 93% title(’b) Contributions to two phase pressure drop’) 94 95% figure(’name’,’Phase Change 3’); 96% subplot(1,2,1) 97% plot(vapor_quality.lng,delta_Pa.lng) 98% grid on 99% xlabel(’Vapor Quality’) 100% ylabel(’Accelerational Pressure drop[Pa]’) 101% title(’a) Two phase pressure drop’) 102% subplot(1,2,2) 103% plot(vapor_quality.lng,delta_Pf.lng) 104% grid on 105% xlabel(’Vapor Quality’) 106% ylabel(’Frictional Pressure drop[Pa]’) 107% title(’b) Two phase pressure drop’) 108 109 110 111%% Heat Flux 112 113 Superheattemp=Walltemp_tp T_lng_tp ; − 114 figure(’name’,’Two phase heat flux’); 115 ax = plotyy(vapor_quality.lng ,q_tp2/1000,vapor_quality.lng ,Superheattemp); 116 grid on 117 xlabel(’Vapor quality’) 118 set(get(ax(1) ,’Ylabel’),’String’,’Heat flux[kW/m^2]’) 119 set(get(ax(2) ,’Ylabel’),’String’,’Superheat temperature[K]’) 120 title(’Two phase heat flux and superheat temperature’) 121 122%% FLow Properties 123 figure(’name’,’Flow Properties’); 124 subplot(2,2,1) 125 plot(Total_Length ,Reynolds_air_final ,co,Relng_sp1,co(end)+(1:Counter) *dx,Relng_tp_liq ,co(end)+(1:Counter) * dx,Relng_tp_gas ,co(end)+Counter *dx+counter2 *dx,Relng_sp2) 126 grid on 127 xlabel(’Length[m]’) 128 ylabel(’Reynolds number’) 124 C.C ODE

129 legend(’Air’,’LNG_{SP1}’,’LNG_{TP,Liq}’,’LNG_{TP,Gas}’,’LNG_{SP2}’,’Location’,’NorthWest’) 130 title(’a) Reynolds number distribution’) 131 132 subplot(2,2,2) 133 plot(Total_Length , Prandtl_air_final ,co,Prlng_sp1 ,co(end)+(1:Counter) *dx,Pr_lng_tp ,co(end)+Counter *dx+ counter2 *dx,Prlng_sp2) 134 grid on 135 xlabel(’Length[m]’) 136 ylabel(’Prandtl number’) 137 legend(’Air’,’LNG_{SP1}’,’LNG_{TP}’,’LNG_{SP2}’,’Location’,’NorthWest’) 138 title(’b) Prandtl number distribution’) 139 140 subplot(2,2,3) 141 plot(Total_Length , v_air_tot ,Total_Length ,v_lng_tot) 142 grid on 143 xlabel(’Length[m]’) 144 ylabel(’Flow speed velocity[m/s]’) 145 legend(’Air’,’LNG’,’Location’,’NorthWest’) 146 title(’c) Flow velocity distribution’) 147 148 subplot(2,2,4) 149 plot(Total_Length ,rho_air_tot ,Total_Length ,rho_lng_tot ,(co(end)+(1:Counter) *dx) ,density_gas) 150 grid on 151 xlabel(’Length[m]’) 152 ylabel(’Density[kg/m^3]’) 153 legend(’Air’,’LNG’,’LNG gas’) 154 title(’d) Density distribution’) 155% subplot(2,4,7) 156% plot(co,visc_lng_sp1) 157% xlabel(’LengthX[m] ’ ) 158% ylabel(’my_{LNG}’) 159% subplot(2,4,8) 160% plot(co,ktlng_sp1) 161% xlabel(’LengthX[m] ’ ) 162% ylabel(’Thermal conductivity LNG[W/(m.K)]’) 163 164 165%% Fin Temperature 166 figure(’name’,’Fin Temperature Distribution along the HEx’); 167 [X Y] = meshgrid(Total_Length ,length_fin) ; 168 surf(X,Y *1000,Total_fin_t) 169 shading interp 170 ylabel(’Fin Length[mm]’) 171 xlabel(’Heat exchanger length[m]’) 172 zlabel(’Fin temperature[K]’) 173 title(’Fin temperature distribution’)

C.29. PRESSUREPLOT

Program/Pressureplot.m 1 pressuredrop_air(1)=delta_P_tot_air(1); 2 for i=2:length(delta_P_tot_air) 3 pressuredrop_air(i)=delta_P_tot_air(i)+pressuredrop_air(i 1) ; − 4 end 5 6 pressuredrop_lng(1)=delta_P_tot_lng(1) ; 7 for i=2:length(delta_P_tot_lng) 8 pressuredrop_lng(i)=delta_P_tot_lng(i)+pressuredrop_lng(i 1) ; − 9 end 10 11 12%[ax,H1,H2]= plotyy(Total_Length, pressuredrop_air, Total_Length, pressuredrop_lng); 13% set(ax,’fontsize’,15) 14% grid on 15% xlabel(’Length[m] ’ ) 16% set(get(ax(2),’Ylabel’),’String’,’LNG pressure drop in[Pa]’,’fontsize’,15) 17% set(get(ax(1),’Ylabel’),’String’,’Air pressure drop in[Pa]’,’fontsize’,15) 18% set(H2,’LineStyle’,’ ’) − − 19% set(ax(1),’YTick’, 1000:1000:4000) − C.30. TP_FRIC_FRAC 125

20% set(ax(2),’YTick’,0:1000:5000) 21 one=plot(Total_Length ,fliplr (pressuredrop_air) ,Total_Length ,pressuredrop_lng ,’ ’); −− 22 set(one,’LineWidth’,3) 23 set(gca,’fontsize’,25) 24 axis([0 2.5 600 4500]) − 25 legend(’Air pressure drop’,’LNG pressure drop’,’Location’,’NorthWest’) 26 grid on 27 title(’a) Pressure drop distribution’) 28 xlabel(’Length[m]’) 29 ylabel(’Pressure drop in[Pa]’,’fontsize’,25) 30%arrow([1 145],[0.5 145])%AIr arrow 31%arrow([0.25 570] ,[0.75 570])%AIr arrow − − 32%% 33 pressuredrop2_lng(1)=delta_Pa.lng(1) ; 34 for i=2:length(delta_Pa.lng) 35 pressuredrop2_lng(i)=delta_Pa.lng(i)+pressuredrop2_lng(i 1) ; − 36 end 37 pressuredrop3_lng(1)=delta_Pf.lng(1) ; 38 for i=2:length(delta_Pf.lng) 39 pressuredrop3_lng(i)=delta_Pf.lng(i)+pressuredrop3_lng(i 1) ; − 40 end 41 figure(’name’,’Phase change2’); 42 subplot(1,2,1) 43 set(gca,’fontsize’,25) 44 one=plot(temp, pressuredrop2_lng+pressuredrop3_lng) ; 45 set(one,’LineWidth’,3) 46%axis([0.8 1.20 5]) 47 grid on 48 xlabel(’Length[m]’) 49 ylabel(’Total pressure drop[Pa]’) 50 title(’a) Total two phase pressure drop’) 51 52 subplot(1,2,2) 53 set(gca,’fontsize’,25) 54 one=plot(temp,pressuredrop2_lng ,temp,pressuredrop3_lng ,’ ’); −− 55 set(one,’LineWidth’,3) 56%axis([0.8 1.20 5]) 57 grid on 58 xlabel(’Length[m]’) 59 ylabel(’Pressure drop[Pa]’) 60 legend(’Accelerational’,’Frictional’,’Location’,’SouthEast’) 61 title(’b) Contributions to two phase pressure drop’)

C.30. TP_FRIC_FRAC

Program/tp_fric_fac.m 1%function to calculate the two phase friction factor according to the model 2%of mueller steinhagen and heck. 3 4%INPUTS:Reynolds number 5%OUTPUTS:friction factor 6 7 function[f]=tp_fric_fac(Re) 8 if isnan(Re) ==1 9 f =0; 10 disp(’Reynoldsnumber nota number’) 11 elseif Re <= 1187 12 f =64/Re ; 13 elseif Re > 1187 14 f =0.3164/Re^0.25; 15 else disp(’Error: Reynolds number out of Bounds tp_fric_fac’) 16 disp([’Re=’ num2str(Re)]) 17 return 18 end

C.31. TP_HEAT

Program/tp_heat.m 126 C.C ODE

1%Function to determine the two phase heat transfer 2 3%INPUTS:vapor quality, wall temperature, critical pressure, pressure LNG, 4%temperature LNG, inner radius, mass flow LNG, number of tubes, Counter, 5%molecular weight, step size dx 6%OUTPUTS: two phase heat transfer coefficient, heat flux, flow speed LNG 7 8 function[h_TP,q,v_lng]=tp_heat(x_dot ,T_w,Pc,P_lng ,T_lng , r_i ,mdot_lng,No_tubes,Counter,M,dx) 9%Precalculations: 10 X_ave=(x_dot(Counter)+x_dot(Counter 1) ) /2;%average vapor quality in the last section(perhaps better the − l a s t final value? 11 delta_Ts=T_w T_lng ;%saturation temperature − 12 p_r=P_lng*100000/Pc.lng;%reduced pressure(dimesionless) 13 Pr_l=cp_lng_liq(T_lng,P_lng) * my_lng_liq(T_lng,P_lng)/kt_lng_liq(T_lng,P_lng) ; 14%v_lng=mdot_lng/(pi r _ i^2 No_tubes)/(rho_lng_gas(T_lng,P_lng) x_dot(end)+rho_lng_liq(T_lng,P_lng) abs(1 * * * * − x_dot(end)));% avg flow speed per tube 15 v_lng=mdot_lng /(pi * r _ i ^2*No_tubes)/(rho_lng_liq(T_lng,P_lng)) ;%flow speed per tube(all as liq), the higher the flow speed, the longer the tp part endures 16 Re_L=rho_lng_liq(T_lng,P_lng) * v_lng *2* r_i/my_lng(T_lng ,P_lng) ;% Reynolds number under consideration of liquid phase only 17 18%Calculation of parameters: 19 M=M. lng *1000;% molar mass; conversion from kg/mol tog/mol 20 A_p=55 p_r^0.12 ( log10(p_r))^( 0.55) M^( 0.5) ;%pool boiling * * − − * − 21 F=(1+X_ave Pr_l (rho_lng_liq(T_lng ,P_lng)/rho_lng_gas(T_lng ,P_lng) 1) ) ^0.35;%Enhancement factor * * − 22 S=(1+0.055 F^0.1 Re_L^0.16)^( 1) ;%suppression factor * * − 23 h_L=0.023*Re_L^0.8* Pr_l ^0.4* kt_lng_liq(T_lng,P_lng)/(2* r _ i ) ;%forced convection dittus boelter heat transfer coefficient 24 25%Coefficients of the cubic equation: 26 f i r s t =1; 27 second= 1 ((S A_p/(F h_L) ) ^2 (F h_L delta_Ts)^(4/3)); − * * * * * * 28 third =0; 29 fourth= 1; − 30 31 solution=roots([first second third fourth]);%finding heat flux in Liu Winterton 32 q=solution (imag(solution)==0);% Take only the real roots in Watt/m2 33 34 h_pool=55 p_r ^0.12 q^(2/3) ( log10(p_r))^ 0.55 M^ 0.5; * * * − − * − 35 h_TP=sqrt((F *h_L) ^2+(S*h_pool)^2); 36 37 38 39%% Kandlikar Correlation 40 41%G=mdot_lng/No_tubes/(r_i^2 * pi);%Mass flux per tube in kg/m2/s 42% d_hyd=2 * r _ i;%calculation of hydraulic diameter 43% h_l=0.023 *Re_L^0.8 * Pr_l^0.4 * kt_lng_liq(T_lng,P_lng)/d_hyd;%single phase heat transfer dittus boelter 44% Bo=q/(G * heatofvap(T_lng));%Boiling number 45% Co=((1 x_dot)/x_dot)^0.8 (rho_lng_gas(T_lng,P_lng)/rho_lng_liq(T_lng,P_lng))^0.5;%Convection number − * 46% Fr_lo=G^2/(rho_lng_liq(T_lng,P_lng)^2 *2* r _ i *9.80665);%Froude number 47% F_fl=1.3;%Fluid factor 48% 49% C1=1.136; 50% C2= 0.9; − 51% C3=667.2; 52% C4=0.7; 53% if Fr_lo >0.04 54% C5=5; 55% else C5=0.3; 56% end 57% 58% h_TP1=(C1 *Co^C2*(25* Fr_lo)^C5+C3 *Bo^C4* F_fl) * h_l; 59% 60% C1=0.6683; 61% C2= 0.2; − 62% C3=1058; 63% C4=0.7; 64% if Fr_lo >0.04 65% C5=5; 66% else C5=0.3; C.32. VAPOR_QUALITY 127

67% end 68% 69% h_TP2=(C1 *Co^C2*(25* Fr_lo)^C5+C3 *Bo^C4* F_fl) * h_l; 70% h_TP=max(h_TP1,h_TP2); 71 72%% Gunger and Winterton 73% if x_dot(end)>=1 74% x_dot=0.99 75% end 76%G=mdot_lng/No_tubes/(r_i^2 * pi);%Mass flux per tube in kg/m2/s 77% p_r=P_lng *100000/Pc.lng;%reduced pressure(dimesionless) 78% Bo=q/(G * heatofvap(T_lng));%Boiling number 79% Re_l=G (1 x_dot(end)) 2 r _ i/my_lng_liq(T_lng,P_lng);%Lqiuid share reynolds number * − * * 80% Pr_l=cp_lng_liq(T_lng,P_lng) * my_lng_liq(T_lng,P_lng)/kt_lng_liq(T_lng,P_lng);%Liquid share Prandlt number 81% Xtt=((1 x_dot(end))/x_dot(end))^0.9 (rho_lng_gas(T_lng,P_lng)/rho_lng_liq(T_lng,P_lng))^0.5 (my_lng_liq( − * * T_lng,P_lng)/my_lng_gas(T_lng,P_lng))^0.1;%Lockart Martinelli parameter 82% Fr=G^2/(rho_lng_liq(T_lng,P_lng)^2 *2* r _ i *9.80665);%Froude number 83% if Fr<0.05 84%E=(1+24000 Bo^1.16+1.37 (1/Xtt)^0.86) Fr^(0.1 2 Fr); * * * − * 85%S=(1+1.15 10^ 6 E^2 Re_l^1.17)^ 1 sqrt(Fr); * − * * − * 86% else 87%E=1+24000 *Bo^1.16+1.37 *(1/Xtt)^0.86 88%S=(1+1.15 10^ 6 E^2 Re_l^1.17)^ 1 * − * * − 89% end 90% 91% h_l=0.023 * Re_l^0.8 * Pr_l^0.4 * kt_lng_liq(T_lng,P_lng)/(2 * r _ i) 92% h_pool=55 p_r^0.12 ( log10(p_r))^( 0.55) M^( 0.5) q^0.67 * * − − * − * 93% h_TP=E * h_l+S *h_pool 94 95%% Bertsch correlation 96% D_h=2 * r _ i; 97%G=mdot_lng/No_tubes/(r_i^2 * pi);%Mass flux per tube in kg/m2/s 98% p_r=P_lng *100000/Pc.lng;%reduced pressure(dimesionless) 99% sigma=surf_tens_lng(T_lng,P_lng) 100% Co=sqrt(sigma/(9.80665 (rho_lng_liq(T_lng,P_lng) rho_lng_gas(T_lng,P_lng)) D_h^2)); * − * 101% 102% Re_l=G *D_h/my_lng_liq(T_lng,P_lng) 103% Re_g=G *D_h/my_lng_gas(T_lng,P_lng) 104% Pr_l=cp_lng_liq(T_lng,P_lng) * my_lng_liq(T_lng,P_lng)/kt_lng_liq(T_lng,P_lng);%Liquid share Prandlt number 105% Pr_g=cp_lng_gas(T_lng,P_lng) *my_lng_gas(T_lng,P_lng)/kt_lng_gas(T_lng,P_lng);%Liquid share Prandlt number 106% 107% h_conv_l=(3.66+(0.0668 *D_h/dx * Re_l * Pr_l)/(1+0.04 *(D_h/dx * Re_l * Pr_l)^(2/3))) * kt_lng_liq(T_lng,P_lng)/D_h; %liquid share convective heat transfer coefficient 108% h_conv_g=(3.66+(0.0668 *D_h/dx *Re_g* Pr_g)/(1+0.04 *(D_h/dx *Re_g* Pr_g)^(2/3))) * kt_lng_gas(T_lng,P_lng)/D_h; %vapor share convective heat transfer coefficient 109% 110% h_conv=h_conv_l (1 x_dot(end))+h_conv_g x_dot(end); * − * 111% h_nb=55 p_r^0.12 ( log10(p_r))^( 0.55) M^( 0.5) q^0.67; * * − − * − * 112% 113%S=1 x_dot(end); − 114%F=1+80 exp( 0.6 Co) (x_dot(end)^2 x_dot(end)^6) * − * * − 115% 116% h_TP=h_nb *S+h_conv *F

C.32. VAPOR_QUALITY

Program/vapor_quality.m 1%% Function to determine the vapor quality 2 3%INPUTS: vapor quality old section, heat flux, latent heat of vaporization, 4%gas mass, total mass 5%OUTPUTS:new vapor quality, new gas mass, total mass 6 7 function[x_dot ,M_gas,M_tot]=vapor_quality(x_dot ,Q, delta_h_lg , M_gas,M_tot) 8 if x_dot == 0 9 M_gas=Q/ delta_h_lg ;%Gas mass calculation if only liquid (=first time) 128 C.C ODE

10 if M_gas > M_tot .sec%Check whether produced gas is more than available liquid 11 disp(’Error: Heat influx too high: vapor quality’) 12 end 13 x_dot=M_gas/M_tot .sec;%calcualte vapor quality 14 else 15 M_gas_new=Q/ delta_h_lg ;%calculate new"piece" of mass of gas 16 if M_gas_new > M_tot .sec 17 disp(’Error: Heat influx too high: vapor quality’) 18 end 19 M_gas=M_gas+M_gas_new ;%Add new piece of gas to old one 20 x_dot=M_gas/M_tot .sec;%calcualte new vapor quality 21 end 22 if x_dot > 1%set erroreous vapor quality to1 23 x_dot = 1; 24 end

C.33. VAPOR_QUALITY_TOOL

Program/vapor_quality_tool.m 1%function to determine the vapor quality ofa section for the mixture of 2%LNG. 3%INPUTS: TemperatureK, Pressure bar, inner tube radiusm, Number of tubes, 4%step, mass flow rate of lng kg/s, lng composition( row vector, length 6), 5%heat inflow 6%OUTPUTS:individual vapor quality, individual vapor mass kg, total mass of 7%substance kg, total vapor quality 8 9 function[x_dot, M_gas, M_tot, alfa ,delta_h_lg ,M_liquid,rho,rho_gas]=vapor_quality_tool(Q,x_dot,M_tot,T,P) 10 11%%Introduce latent heats of vaporization 12 delta_h_lg=heatofvap(T);%in kJ/kg 13 14%%Determine mass of gas 15 M_gas=x_dot *M_tot .sec; 16 17%%Calculate liquid masses and volumes for determining heat influx 18 M_liquid.mass=M_tot.sec M_gas ; − 19 M_liquid.vol=M_liquid.mass/rho_lng_liq(T,P) ; 20 21%%Calculate vapor quality 22 [x_dot ,M_gas,M_tot]=vapor_quality(x_dot ,Q,delta_h_lg ,M_gas,M_tot) ; 23 rho=rho_lng_liq(T,P); 24 rho_gas=rho_lng_gas(T,P) ; 25 alfa=void_fraction(x_dot,rho,rho_gas);

C.34. VOID_FRACTION

Program/void_fraction.m 1% Function to determine the void fraction for the pressure drop 2% INPUT:vapor quality, density of liquid and gas 3% OUTPUT: void fraction of the mixture 4 5 function[alfa]=void_fraction(x_dot, rho, rho_gas) 6 v_gas=1/rho_gas;%determine specific volume of gas 7 v=1/rho ;%determine specific volume of liquid 8 alfa=(1+(1 x_dot)/x_dot (v/v_gas)^(2/3))^ 1;%calculate void fraction − * −

C.35. WALLT

Program/wallT.m 1%Function to estimate the wall temperature according toa paper of Gorman, 2%Sparrow and Abraham 3 4%INPUTS: Air temperature, LNG temperature, Prandtl number LNG, Prandtl 5%number Air 6%OUTPUTS:Wall temperature C.35. WALLT 129

7 8 function T_w = wallT(T_air ,T_lng,Pr_lng,Pr_air) 9 10%Determine outside Tube wall temperature 11 theta_crit =0.784* Pr_lng^(0.0358); 12 corr=(1 theta_crit)/theta_crit; − 13 T_w=( T_lng+T_air * corr)/(corr+1); 14 15%Determin inside Tube wall temperature 16% theta_crit=0.784 * Pr_air^(0.0358); 17% corr=(1 theta_crit)/theta_crit; − 18% T_w=(T_air+T_lng * corr)/(corr+1); 19 20%T_w=(T_w_in+T_w_out)/2%mean wall temperature 21 22 23% clear all 24% close all 25% 26% Oc=0; 27% Ic=0; 28% Pr=3 29% for T_lng=120:20:400 30% Oc=Oc+1 31% for T_air=500:10:890 32% Ic=Ic+1 33% theta_crit=0.784 * Pr^(0.0358); 34% corr=(1 theta_crit)/theta_crit; − 35% T_w(Ic,Oc)=(T_lng+T_air * corr)/(corr+1); 36% end 37% Ic=0 38% end 39 40%surf(T_w) 41%xlabel(’Lng Temperature 120+10 *x’) 42%ylabel(’Air Temperature 500+10 *x’) 43%zlabel(’Resulting Wall temperature’)