PhD Thesis FDM

Faculty of Engineering

MODELLING AND SIMULATION OF

LIQUID ROCKET ENGINE IGNITION

TRANSIENTS

a Dissertation submitted to the Doctoral Committee

of

Tecnologia Aeronautica e Spaziale

in partial fulfilment of the requirements for the

degree of Doctor of Philosophy

Tutor Candidate

Prof. Marcello Onofri Francesco Di Matteo

Co-tutor

Ing. Marco De Rosa

Academic Year 2010-2011

to my family

Contents

Nomenclature x

1. Introduction 1

1.1. Motivation: what is the ignition transient and why are we inter-

ested in it? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Key challenges in Rocket Engine start-up . . . . . . . . . . . . . . 2

1.3. Main objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4. Organization of this Thesis . . . . . . . . . . . . . . . . . . . . . . 5

2. State of the Art 8

2.1. Engine cycles and their start-up and shut-down transients . . . . 8

2.1.1. Gas Generator Engine . . . . . . . . . . . . . . . . . . . . 8

2.1.2. Expander Engine . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3. Staged Combustion Engine . . . . . . . . . . . . . . . . . 12

2.2. Modelling: review of previous works . . . . . . . . . . . . . . . . . 15

3. ESPSS: European Space Propulsion System Simulation 19

3.1. Fluid Properties Library . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1. Perfect Gas properties according to CEA . . . . . . . . . . 21

3.1.2. Perfect Gas interpolated properties . . . . . . . . . . . . . 23

3.1.3. SimpliVed Liquid interpolated properties . . . . . . . . . . 24

3.1.4. Real Fluids interpolated properties . . . . . . . . . . . . . 25

3.1.5. Perfect gas mixtures . . . . . . . . . . . . . . . . . . . . . 27

3.1.6. Real Fluid - Perfect gas mixtures . . . . . . . . . . . . . . 28

3.2. Fluid Flow 1D Library . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1. Components ClassiVcation . . . . . . . . . . . . . . . . . . 33

3.2.2. Junction/Valve . . . . . . . . . . . . . . . . . . . . . . . . . 35

i

Contents

3.2.3. Capacity/Volume . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.4. Tubes/Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3. Turbomachinery Library . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1. Pump & Generic Pump . . . . . . . . . . . . . . . . . . . . 45

3.3.2. Turbine & Generic Turbine . . . . . . . . . . . . . . . . . 47

3.4. Combustion Chambers Library . . . . . . . . . . . . . . . . . . . . 49

3.4.1. Injector Cavity . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2. Combustor Equilibrium . . . . . . . . . . . . . . . . . . . 53

3.4.3. Combustor rate . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.4. Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.5. Cooling Jacket components . . . . . . . . . . . . . . . . . 70

4. Steady State Library 78

4.1. Components Overview . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2. Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3. The “type” switch . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4. 1-D pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5. 0-D components: junctions & valves . . . . . . . . . . . . . . . . . 83

4.6. Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7. Cooling Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.8. Turbomachinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8.1. Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8.2. Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.9. Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.9.1. Component validations . . . . . . . . . . . . . . . . . . . . 96

4.9.2. Subsystem validations . . . . . . . . . . . . . . . . . . . . 100

4.9.3. Engine cycle designs . . . . . . . . . . . . . . . . . . . . . 105

5. Transient Modelling 114

5.1. Injector Plate model . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.1.1. Qualitative behaviour . . . . . . . . . . . . . . . . . . . . 118

5.2. Hot Gas side heat transfer coeXcient models . . . . . . . . . . . . 120

5.2.1. Models implemented . . . . . . . . . . . . . . . . . . . . . 120

5.2.2. Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

ii

Contents

5.3. Q-2D stratiVcation model for HARCC . . . . . . . . . . . . . . . . 126

5.3.1. Model description . . . . . . . . . . . . . . . . . . . . . . . 126

5.3.2. Numerical validation . . . . . . . . . . . . . . . . . . . . . 131

5.3.3. Experimental validation . . . . . . . . . . . . . . . . . . . 132

6. Integrated Validation: RL-10 design and analysis 142

6.1. Overview of the RL-10A-3-3A rocket engine . . . . . . . . . . . . 143

6.2. Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2.1. Turbomachinery modelling . . . . . . . . . . . . . . . . . 147

6.2.2. Thrust chamber and cooling jacket modelling . . . . . . . 152

6.2.3. Lines, valves and manifolds modelling . . . . . . . . . . . 158

6.3. Subsystem simulation: validation at nominal conditions . . . . . . 162

6.4. RL-10 Engine start-up . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.4.1. Description of the start-up sequences . . . . . . . . . . . . 165

6.4.2. Start transient . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.5. RL-10 engine shut-down . . . . . . . . . . . . . . . . . . . . . . . . 175

6.5.1. Description of the shut-down sequence . . . . . . . . . . 175

6.5.2. Shut-down transient . . . . . . . . . . . . . . . . . . . . . 177

6.6. Dynamic Response Analysis . . . . . . . . . . . . . . . . . . . . . 184

7. Conclusions 197

A. Implementation of Up-wind Roe Scheme b

A.1. Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . b

A.1.1. 4-equation subset . . . . . . . . . . . . . . . . . . . . . . . c

A.2. Numerical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . d

A.2.1. Roe’s numerical scheme . . . . . . . . . . . . . . . . . . . d

A.2.2. Approximate Riemann Solver . . . . . . . . . . . . . . . . e

A.3. Reconstruction method . . . . . . . . . . . . . . . . . . . . . . . . i

A.3.1. Higher order accuracy . . . . . . . . . . . . . . . . . . . . i

A.3.2. Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . o

A.3.3. Variable cross-section . . . . . . . . . . . . . . . . . . . . . p

B. Friction Factor Correlations r

iii

Contents

B.1. Single-Phase Friction Factor Calculation. Function hdc_fric . . . r

B.2. Two-Phase Friction Factor Calculation. Friedel Correlation . . . . r

B.3. Elbow Pressure Loss Function . . . . . . . . . . . . . . . . . . . . . s

C. Film CoeXcient Calculation v

iv

List of Figures

2.1. Vulcain 2 schematic [46] . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2. Vinci schematic [45] . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3. Space Shuttle Main Engine schematic [149] . . . . . . . . . . . . . 13

2.4. Space Shuttle Main Engine start-up sequence [12] . . . . . . . . . 14

2.5. Space Shuttle Main Engine shut-down sequence [12] . . . . . . . 15

3.1. Components in the fluid_flow_1d library . . . . . . . . . . . . . 34

3.2. Pipe discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3. Components in the turbo_machinery library . . . . . . . . . . . 45

3.4. Components in the comb_chambers library . . . . . . . . . . . . 52

3.5. Cooling jacket wall mesh [43] . . . . . . . . . . . . . . . . . . . . . 71

3.6. SimpliVed Cooling Jacket wall disposition [43] . . . . . . . . . . . 74

3.7. Channel with relevant areas and surfaces for heat Wux calculation 75

3.8. Longitudinal heat Wuxes for a segment i . . . . . . . . . . . . . . . 77

4.1. Components in the Steady State library . . . . . . . . . . . . . . . 79

4.2. Cooling jacket channels wall mesh [43] . . . . . . . . . . . . . . . 91

4.3. Schematic of the Pipeline test case. Purple: steady state compo-

nents. Cyan: transient components . . . . . . . . . . . . . . . . . . 97

4.4. Schematic of Combustion Chamber test case. Purple: steady state

components. Cyan: transient components . . . . . . . . . . . . . . 99

4.5. Turbopump test case: HM7B power pack transient schematic . . . 102

4.6. Turbopump test case: HM7B power pack steady state schematic . 102

4.7. Chamber test case: HM7B Combustion Chamber transient schematic 103

4.8. Chamber test case: HM7B Combustion Chamber steady state

schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.9. HM7B engine system schematic . . . . . . . . . . . . . . . . . . . 106

v

List of Figures

4.10. Schematic of the RL-10 engine . . . . . . . . . . . . . . . . . . . . 111

5.1. Schematic illustration of an arbitrary injector head . . . . . . . . 115

5.2. Schematics of the injector plates . . . . . . . . . . . . . . . . . . . 116

5.3. Temperature proVles from original and new model . . . . . . . . . 119

5.4. Heat Wuxes and wall temperatures results . . . . . . . . . . . . . . 124

5.5. left: 1-D Wuid element and energy balance used for conventional 1-

D method; right: control volumes of the Q-2D approach integrated

in 3D wall elements . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.6. Cooling jacket wall mesh . . . . . . . . . . . . . . . . . . . . . . . 130

5.7. Methane bulk variables evolution along channel axis . . . . . . . 133

5.8. Design of the 4 sector HARCC segment . . . . . . . . . . . . . . . 134

5.9. Schematic of the experimental test case . . . . . . . . . . . . . . . 136

5.10. Wall and Wuid thermal stratiVcation,AR = 9.2, pc = 88 bar . . . . 138

5.11. Wall and Wuid thermal stratiVcation, AR = 9.2, pc = 58 bar . . . . 139

5.12. Wall and Wuid thermal stratiVcation, AR = 30, pc = 88 bar . . . . 140

5.13. Wall and Wuid thermal stratiVcation, AR = 30, pc = 58 bar . . . . 141

6.1. RL-10A-3-3A engine schematic [115] . . . . . . . . . . . . . . . . . 145

6.2. RL-10A-3-3A engine diagram . . . . . . . . . . . . . . . . . . . . . 147

6.3. Pumps performance maps . . . . . . . . . . . . . . . . . . . . . . . 149

6.4. Iterative procedure for determining pump parameters . . . . . . . 150

6.5. Turbine performance maps from P&W [15] . . . . . . . . . . . . . 151

6.6. Turbine performance maps . . . . . . . . . . . . . . . . . . . . . . 152

6.7. RL-10A-3-3A chamber contour [15] and discretisation . . . . . . . 154

6.8. Cooling jacket channels proVles . . . . . . . . . . . . . . . . . . . 157

6.9. Venturi nozzle proVle . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.10. RL-10A-3-3A schematic model . . . . . . . . . . . . . . . . . . . . 164

6.11. RL-10A-3-3A Valve schedule for Start-up Simulation [15] . . . . . 166

6.12. Valves opening sequence adopted in the simulation . . . . . . . . 168

6.13. Transient results - part 1 . . . . . . . . . . . . . . . . . . . . . . . . 171




vi

List of Figures

6.17. RL-10A-3-3A Valve schedule for Shut-down Simulation [15] . . . 176

6.18. Valves closing sequence adopted in the simulation . . . . . . . . . 177

6.19. Shut-down results - part 1 . . . . . . . . . . . . . . . . . . . . . . . 180




6.23. TCV throttle results - part 1 . . . . . . . . . . . . . . . . . . . . . . 187




6.27. OCV throttle results - part 1 . . . . . . . . . . . . . . . . . . . . . . 193




A.1. Piece-wise linear reconstruction. . . . . . . . . . . . . . . . . . . . l

B.1. Elbow pressure loss parameters . . . . . . . . . . . . . . . . . . . . t

vii

List of Tables

4.1. 1-D pipe element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2. 0-D Junction element . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3. Combustor element . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4. Nozzle element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5. Cooling jacket element . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6. Regenerative circuit element . . . . . . . . . . . . . . . . . . . . . 90

4.7. Pump element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.8. Turbine element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.9. Pipeline input data . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.10. Pipeline output data . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.11. CC input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.12. CC output data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.13. HM7B Turbopump input [44] and initial data . . . . . . . . . . . . 101

4.14. HM7B Turbopump output data . . . . . . . . . . . . . . . . . . . . 101

4.15. HM7B CC input [44] and initial data . . . . . . . . . . . . . . . . . 104

4.16. HM7B CC output data . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.17. HM7B input [44] and initial data . . . . . . . . . . . . . . . . . . . 107

4.18. HM7B engine system output data . . . . . . . . . . . . . . . . . . 108

4.19. RL-10A-3-3A input and initial data . . . . . . . . . . . . . . . . . . 110

4.20. RL-10A-3-3A engine system output data . . . . . . . . . . . . . . . 112

5.1. Injector plate variables comparison . . . . . . . . . . . . . . . . . . 119

5.2. Cooling channels geometries . . . . . . . . . . . . . . . . . . . . . 134

5.3. Positioning of themocouples . . . . . . . . . . . . . . . . . . . . . 135

6.1. RL-10A-3-3A construction data [15] . . . . . . . . . . . . . . . . . 146

6.2. Venturi geometrical data . . . . . . . . . . . . . . . . . . . . . . . . 159

viii

List of Tables

6.3. Fuel line valves parameters . . . . . . . . . . . . . . . . . . . . . . 160

6.4. Oxidiser line valves parameters . . . . . . . . . . . . . . . . . . . . 161

6.5. RL-10A-3-3A engine system output data . . . . . . . . . . . . . . . 163

6.6. Engine dynamic response to TCV ±10% operation . . . . . . . . . 186

6.7. Engine dynamic response to OCV ±10% operation . . . . . . . . . 192

A.1. DiUerent values of ω. . . . . . . . . . . . . . . . . . . . . . . . . . . k

ix

Nomenclature

A Cross section area

AR Aspect ratio

C Capacitive

C+ Reduced torque

Cp SpeciVc heat

D Diameter

E Internal energy

fr Friction factor

G Mass Wow per unit area

h, H Enthalpy

h Dimensionless characteristic head

hc Heat transfer coeXcient

Isp SpeciVc impulse

K Loss coeXcient for design condition

k Concentrated load losses

L Characteristic length

M Mass

M Mach number

MR Mixture ratio

MW Molar weight

mh Enthalpy Wow

N Speed coeXcient

Nk Number of moles

Ns SpeciVc speed

n Reduced speed

ns Pump number of suctions

nst Pump number of stages

P Pressure

P Perimeter

Pr Prandtl number

Q Volumetric Wow rate

Q+ Mass Wow coeXcient

Q Heat Wux

q Heat Wux per unit area

R, r Geometrical radius

R Universal gas constant

Re Reynolds number

S Entropy

St Stanton number

T Temperature

TDH Total dynamic head

t Wall thickness

u Internal energy

u Primitive variables vector

V Volume

v Velocity

W Turbopump power

We Weber number

x Fluid quality

x Horizontal abscissa

xnc Non-condensable mass fraction

Z Compressibility factor

x

Nomenclature

Greek symbols

α Fluid void fraction

β Volumetric expansivity

β Dimensionless characteristics torque

γ Variable isentropic coeXcient

δ total to static pressure ratio

ε Absolute roughness

ζ Concentrated pressure drop parameter

η EXciency

ϑ Pump dimensionless parameter

θ Total to static temperature ratio

κ Isothermal compressibility

λ Thermal conductivity

µ Fluid viscosity

ν SpeciVc volume

ν Reduced Wow parameter

ξ Pipe pressure drop coeXcient

Π Total to total pressure ratio

ρ Density

σ Stefan-Boltzmann constant

τ Mechanical torque

τ Time constant

τxy Fluid shear stress

φ Interaction parameter

φ+ Mass Wow coeXcient

ψ+ Head rise coeXcient

Ω Acentric factor

ω Rotational speed

Subscripts

aw Adiabatic wall

amb Ambient

bu Burned condition

c Fluid critical point

c, cc Combustion chamber

ch Channel

cond Conductive

cap Capacitive

conv Convective

cav Cavity

crit Critical condition

chem Chemical composition vector

eff EUective

ext External

eq Equilibrium condition

fu Fuel

fr Frozen condition

g, gas Gaseous phase

gg Gas generator

hg Hot gas side

i, j, k Spatial indices

i Internal wall condition

jun Junction

l, liq Liquid phase

mix Mixing condition

nc Non-condensable fraction

o Initial condition

ox Oxidiser

p Pump

pw Powder

R Rated condition

rad Radiative

ref Reference point

sat Saturated condition

sound Sound condition

t Turbine

t Turbulent

xi

Nomenclature

th Throat

v, vap Vapour

w Wall

wet Wet

xii

1. Introduction

In this introductory chapter the problem of the ignition start-up and shut-down

transients for liquid rocket engine is discussed. The object of this thesis is not

limited to the modelling and simulation of liquid rocket engines start-up but it

aims to the creation of a tool able to model the dynamic behaviour and to obtain

the functional design of liquid rocket engine systems.

Once the motivation and the interest of this problem are shown, the attention

will be focused on the critical aspects that characterize liquid rocket engine

propulsion system during these particular phases. Finally, the main objectives of

this dissertation as well as a brief overview of its structure will be shown.

1.1. Motivation: what is the ignition transient and why are we

interested in it?

In the past, until the end of the 70s, the starting process development of liquid-

propellant engines was usually achieved empirically by testing diUerent schemes

and start cyclogrammes directly during test Vring. It required a big amount of

resources and was time consuming. Start-up calculation methods at that time did

not reWect the main factors aUecting the process and could not serve as reliable

means for start development. More than 30% of the engine failures occurred

during start-up [68]. Recently, with more powerful and complex engines, the

need of more rationale and reliable methods of the starting process development

appeared.

The prediction of the start-up characteristics of liquid propellant rocket engines

is important to the engine conVguration and control system design processes.

Despite that, engine start systems have received secondary considerations since

high-power performance was the chief design objective.

1

1. Introduction

A careful synchronization of control actions with transient start processes

is required to deliver the smooth and reliable thrust build-up characteristics

desired. For turbopump-fed engines, the ability to power turbomachinery prior to

combustion chamber ignition is an important design concern [7]. Indeed, thrust

build-up can be delayed or inhibited if turbine power is insuXcient to accelerate

propellant pumps.

Many diXculties associated with engine start predictions stem from the non-

linear mass Wow and heat transfer characteristics associated with Vlling uncondi-

tioned engine systems with cryogenic propellants.

The use of cryogenic Wuids imposes additional problems during start-up when

they Wow into a system with ambient wall temperatures. Because of strong

evaporation inside tubes, cryogenic fuels may lead to severe over-pressures and

even Thermal choking during the cool-down processes. Accurate predictions are

especially important in engine cycles where this initial propellant Wow provides

all of the available turbine power for starting, e.g., expander cycles.

During the start of liquid-propellant engine the main role belongs to the hy-

drodynamic processes (mass Wow variations, pressure surges, phase change, etc.).

Hydrodynamic processes are practically the only ones that can be used to act

upon all other phases of the physical-chemical process forming the start-up of the

engine.

1.2. Key challenges in Rocket Engine start-up

Ever since the Vrst liquid rocket engines were developed, performance analyses

have been implemented to examine their operation under various conditions.

Steady state assessments for design conditions are wide-spread and useful in

pre-design phases [17, 47, 61, 94, 100]. Further testing of the nominal operational

modes are performed during initial testing of new liquid rocket engines when

various tests are run in order to verify that the engine has been constructed

according to the original design and performance requirements. This leads to

engines that are extensively reliable during their nominal operation.

Nominal conditions however do not illustrate the extreme conditions in which

most engine components are required to work during transient phases, such as

2

1. Introduction

start-up, shut-down, and throttling. Transient phenomena range from combustion

high frequency instabilities to water hammer eUects in feed lines. High pressure

and temperature peaks, inherent to transient phases, may lead to failures in part

of the engine system. The potentially resulting system failures may cause loss

of payload, serious damage to the ground segment, if not loss of human life.

Anomalies in the past, such as the RL-10 anomaly of the Atlas-Centaur Wight

AC-71 in 1992 [15], or the Aestus anomaly of Ariane 5G Wight 142 in 2001 [74, 82]

have demonstrated that a hard transient may indeed lead to signiVcant system

failures.

In order to have a better understanding of the main problems that may occur

during start-up and shut-down, a description of the main failures for the Japanese

LE-7 engine and for the American SSME engine are here described.

The Space Shuttle Main Engine (SSME) is the high performance LOX/LH2 engine

which was used in the Space Shuttle, producing a thrust of about 1700 kN by

means of a staged combustion cycle. For this very complex engine 5 years of

analysis were necessary to model the transient behaviour of the propellants and

of the hardware during start-up and shut-down.

The SSME engine was sensitive to small changes to propellant conditions and

valve tuning was critical, a 2% error in the valve position or a 0.1 s timing error

could lead to signiVcant damage to the engine.

A step-by-step approach was necessary to explore the start-up sequence with

small time increments: that required 19 tests, 23 weeks, 8 turbopump replacements

to reach 2 s into 5 s of start-up. Additional 18 tests, 12 weeks and 5 turbo-pump

replacements were necessary to touch the minimum power level [12].

The LE-7 is a high performance LOX/LH2 engine employed in the H-II rocket,

which produces a thrust of about 1000 kN by means of a staged combustion cycle

similar to that of the SSME. During H-II Flight 8 in 1999 a failure occurred in

the LE-7 engine. The failure was determined by mechanical vibration problems

into the fuel pump that caused high cycle fatigue and so the premature engine

failure [129]. During development tests of the LE-7 other issues can be addressed

to phenomena occurring during transient phases [50]:

3

1. Introduction

• High sensitivity to heat transfer from hardware to Wuids and to ignition

timing reproducibility of both combustion devices

• Functional instability to inlet pump vibration of pressure

• Problem of rotating cavitation on oxidiser and fuel pumps

• Over power during start-up caused damages to hardware

• Explosion at Main Oxidiser Valve opening

1.3. Main objectives

Given the increased power of today’s computers, more frequent use is made of

CFD codes to perform detailed assessments of the Wow behaviour in single engine

components. The goal is to understand observed variations in Wuid properties and

to Vnd the source of unexpected behaviours. Such CFD methods however require

extensive computational times, making speedy parametric analyses impossible.

Additionally, most require long input preparation times. Although the qualities

of a thorough and in full-depth analysis may be desirable in the study of speciVc

single components and scenarios, for complex, multi-component systems, the long

computational times become insurmountable.

An intelligent simpliVcation of the underlying processes allows the reduction

of the 3-D governing partial diUerential equations to one-dimensional or quasi

1-D diUerential equations which no longer require complex solution methods thus

allowing much faster computational times. Results obtained from such studies

rely signiVcantly on imposed initial and boundary conditions. SimpliVcations

introduced in these models often lead to an incorrect transient behaviour which

does not correspond to the measured and observed physics. This is, amongst

other reasons, due to the ignored interaction between downstream and upstream

lying components. Tools concentrating on one component only are thus not

suXcient as they do not help in understanding how components aUect each other

during transient phases, what their impact on system frequencies is, and how this

interaction may lead to a major component or system failure.

4

1. Introduction

A system approach is then necessary to take into account all the interactions

between all the components of an engine. This choice is fundamental if a detailed

estimation of the engine transient behaviour is the task.

To this purpose the present work was initiated with the aim of studying and

modelling by numerical tool the ignition transient of liquid rocket engines, improv-

ing and implementing more complex and accurate models in system modelling

tools for transient analysis. This aim is achieved in three steps:

• A suitable physical and mathematical collection of models able to design

and analyse the steady state behaviour of liquid rocket engine systems is

developed and implemented in a numerical code. The model library has

been successfully validated with respect to open literature data, performing

design simulations and oU-design analyses of actual rocket engines.

• Basic and simple models used to simulate the injector plate of the engine,

the hot-gas-side heat transfer correlations in the combustion chamber and

the regenerative circuit are exchanged with improved, more sophisticated

and more physical models. These models are able to describe accurately

the convective and radiative heat transfer at the injector plate face, the

hot-gas-side heat transfer coeXcient, and to describe also the inWuence of

thermal stratiVcation in high aspect ratio cooling channels.

• At last, the validation of a design procedure and the models developed

are achieved by the design, the start-up, the shut-down and the dynamic

response simulation of a real engine, the RL-10A-3-3A.

1.4. Organization of this Thesis

The work performed is here presented in six chapters as described below.

Chapter 1 provides a brief introduction of the motivations which led this work,

followed by a description of the main issues related to transient phases in liquid

rocket engines. Issues presented practical consequence as described for the case of

LE-7 engine failures or the SSME start-up campaign.

5

1. Introduction

In addition, a description of the main objectives of this work have been provided

explaining which approach has been adopted for the modelling of the components’

behaviour of a liquid rocket engine, and what is the Vnal aim of such a Ph.D.

dissertation.

Chapter 2 is dedicated to a summary of the state of the art in the Veld of liquid

rocket engine transient simulation. Models and tools from USA, Europe, Russia,

Japan, Iran, China are collected, studied and compared to each other in order

to Vnd advantages and drawbacks present in each work.The literature study is

fundamental to understand in which direction this work should be directed.

Chapter 3 summarizes the ESPSS library. It describes the models implemented in

the European Space Propulsion System Simulation library, a collection of models

for each engine’s component used as starting point for this work. The chapter

includes the way the library and the main components are modelled: Wuid prop-

erties, pipes, volumes, valves and junctions components, pumps and turbines,

combustion devices, cooling channels and nozzles.

The assumptions behind each formulation as well as the way components inter-

act with each other are analysed; advantages and limitations of each models are

presented in this chapter.

Chapter 4 illustrates the steady state modelling. It examines the most important

engine components models developed for the creation of a steady state library.

The main purpose of these components is the design and the analyses in steady

state conditions liquid rocket engine systems cycles.

The development of these models is used to enhance the system capabilities of

the code and create a fundamental instrument to be used along the entire design

period from pre-design phase to parametric studies for Vne tuning of the engine

parameters.

The Steady State library presented within this chapter enables to perform iter-

ative engine design loops and parametric studies in a reasonable computational

time.

6

1. Introduction

Chapter 5 concerns the transient modelling. It provides a description of all

the new models developed to enhance and improve the simulation capabilities

during transient phases of a propulsion system. Three new models are here im-

plemented and discussed: a new injector plate model, a new formulation for the

evaluation of the heat Wuxes in combustion devices, Vnally a new model for the

evaluation of thermal stratiVcation on cooling channels is described.

For each model a discussion of the derived equations is provided as well as test

cases for validation purposes. Where available models simulations are compared

with numerical test cases or experimental results.

Chapter 6 describes the system level validation. It contains the main test case

examined: the RL-10A-3-3A rocket engine which presents signiVcant challenges

due to its expander cycle conVguration and its ignition settings. Modelling of the

liquid rocket engine in the frame of its transient phases (start-up and shut-down)

and the resulting simulation data are discussed.

Chapter 7 concludes this thesis summarising the main achievements of the work

performed and indicating those points which, in the frame of further work, could

be improved and others which could be considered in a further extension of the

models presented.

7

2. State of the Art

2.1. Engine cycles and their start-up and shut-down transients

An engine cycle for turbopump-fed engines describes the speciVc propellant Wow

paths through the major engine components, the method of providing the hot gas

to one or more turbines, and the method of handling the turbine exhaust gases.

Depending of the propellant path considered we will have (among the main ones)

a gas generator cycle, an expander cycle or a staged combustion cycle.

Each system displays a speciVc sequence for start-up and shut-down phases.

The sequences are tailored to Vt general engine characteristics such as engine

cycle, as well as more speciVc details such as ignition system type. A gas-generator

engine will therefore have a diUerent sequence to an expander cycle or a staged

combustion engine.

2.1.1. Gas Generator Engine

In the gas generator cycle the turbine inlet gas comes from a separate gas generator.

Its propellants can be supplied from separate propellant tanks or can be bled oU

the main propellant feed system. This cycle is relatively simple; the pressures in

the liquid pipes and pumps are relatively low (which reduces inert engine mass).

It has less engine speciVc impulse than an expander cycle or a staged combustion

cycle. The pressure ratio across the turbine is relatively high, but the turbine or

gas generator Wow is small (1 to 4% of total propellant Wow) if compared to closed

cycles.

Alternatively, this turbine exhaust can be aspirated into the main Wow through

openings in the diverging nozzle section. This gas then protects the walls near the

nozzle exit from high temperatures. Both methods can provide a small amount of

additional thrust. The gas generator mixture ratio is usually fuel rich (in some

8

2. State of the Art

engine it is oxidizer rich) so that the gas temperatures are low enough (typically

900 to 1350 K) to allow the use of uncooled turbine blades and uncooled nozzle

exit segments [133, 94].

The European Vulcain and Vulcain 2 (see Figure 2.1) engines are two examples

of gas generator cycle engines. The gas generator in these engines provides hot

gases to power two turbines, one for each propellant pump.

Figure 2.1.: Vulcain 2 schematic [46]

Prior to commencing the start-up sequence the engines undergo a chill-down

phase of all components with the exception of the main propellant valves and gas

generator, to reduce thermal shock eUects when the cold fuel is fed to the engine

at start-up. This phase lasts ca. 2.5 hours. The engines are then started using a

starter to move the turbine in order to provide a minimal amount of power to

the pumps to feed the gas generator. A pyrotechnic igniter is then used to ignite

9

2. State of the Art

the gas generator. Given the now full power available to the turbines to run the

pumps, feeding of the main combustion chamber is possible.

The fuel valves are opened Vrst, the regenerative cooling channels Vll with

fuel which then irrupts into the combustion chamber which, with aid of another

pyrotechnic igniter, is ignited under lean conditions. The combination of the fuel

rich condition at ignition and the cooling performed by the regenerative cooling

system increases the complexity of the ignition process of the main combustion

chamber [91].

2.1.2. Expander Engine

In the expander cycle most of the engine coolant (usually the fuel) is fed to low-

pressure-ratio turbines after having passed through the cooling jacket where it

picked up energy. Part of the coolant, perhaps 5 to 15%, bypasses the turbine

and rejoins the turbine exhaust Wow before the entire coolant Wow is injected

into the engine combustion chamber where it mixes and burns with the oxidizer.

The primary advantages of the expander cycle are good speciVc impulse, engine

simplicity, and relatively low engine mass. In the expander cycle all the propellants

are fully burned in the engine combustion chamber and expanded eXciently in

the engine exhaust nozzle [133, 94].

The American RL-10 engine and the European Vinci engine (currently under

development) represent two examples for this type of cycle (see Figure 2.2).

The latter is one such expander engine which implements a regenerative cooling

system based on hydrogen to cool the combustion chamber. Hot gases generated

in an electric igniter are implemented as a pilot Wame to start the main combustion

chamber in which the temperature progressively increases as stable combustion

is established. The liquid hydrogen from the tanks leaving the cooling system

undergoes a phase change from liquid to supercritical during the ignition sequence

and thus powers up the turbines as it gains velocity and pressure. For these very

reasons, the Vinci engine unlike other European engines implements reaction

rather than impulsive turbines [91, 38].

The use of LH2 and LOX as propellants in the RL-10 engine, makes necessary a

chill-down step to avoid propellant boiling and pump cavitation during operation.

Cavitation must be avoided due to its propensity to cause erosion damage to the

10

2. State of the Art

Figure 2.2.: Vinci schematic [45]

blades. When the propellants are used for chill-down, the pre-start valves are

opened to allow Wow through the pumps and the tank head pressure provides the

driving force. The LOX is sent through its entire circuit and fed into the main

chamber. The LH2, however, is Wowed out through the cooldown valves prior to

going through the regenerative cooling tubes. This will allow the cooling tubes to

remain at a signiVcantly higher temperature for start-up and avoid any possibility

of inadvertent propellant mixing and igniting prior to start-up.

When the engine start command is given, the cooldown valves are closed and

the main fuel valve is opened. This allows for the LH2 to Wow through the

regenerative cooling lines, which heat up and vaporise the fuel. Once the gaseous

hydrogen begins to drive the turbine, the pumps take over providing the pressure

gradient for Wow. The main chamber is ignited using a pilot Wame created from

gaseous propellants tapped oU from the main lines.

One the combustion chamber is lit, additional heat is transferred into hydrogen

Wow form combustion process. This drives the turbine to its nominal power and

eventually the engine reaches a steady state condition.

When the shut-down command is issued, the main fuel shut-oU valve is closed

11

2. State of the Art

tu cut-oU fuel Wow into the main chamber. As a result, combustion in the combus-

tion chamber and thrust are rapidly terminated. The prestart valves are closed

to stop the propellant Wow and the cooldown valves are opened to prevent over

pressure of the engine due to trapped hydrogen.

2.1.3. Staged Combustion Engine

In the staged combustion cycle, the coolant Wow path through the cooling jacket

is the same as that of the expander cycle. Here a high-pressure precombustor

(gas generator) burns all the fuel with part of the oxidizer to provide high-energy

gas to the turbines. The total turbine exhaust gas Wow is injected into the main

combustion chamber where it burns with the remaining oxidizer. This cycle lends

itself to high-chamber-pressure operation, which allows a small thrust chamber

size. The extra pressure drop in the precombustor and turbines causes the pump

discharge pressures of both the fuel and the oxidizer to be higher than with

open cycles, requiring heavier and more complex pumps, turbines, and piping.

The turbine Wow is relatively high and the turbine pressure drop is low, when

compared to an open cycle. The staged combustion cycle gives the highest speciVc

impulse, but it is more complex and heavy [133, 94].

A variation of the staged combustion cycle is used in the Space Shuttle Main

Engine (SSME) (see schematic in Figure 2.3).

The engine assembly consists of the engine powerhead and main combustion

chamber/nozzle assembly. The powerhead uses two preburners, a main injector,

and an oxidizer heat exchanger, all welded into the hot gas manifold. The preburn-

ers generate fuel-rich combustion gases to drive the LOX and LH2 turbopumps.

Hydrogen fuel is also used to cool the main chamber and nozzle.

Prior to starting the SSME engine, the start preparation phase takes place. This

consists of purging and thermal conditioning followed again by purging. During

the Vrst purging phase, dry nitrogen and dry helium are used to remove moisture

as well as air which would otherwise freeze along the oxidiser and fuel lines

respectively. After thorough purging has been performed, thermal conditioning is

undertaken by allowing propellants to Wow into the engine down to the main fuel

valve (MFV) on the LH2 side and down to three oxidiser valves on the LOX side,

i.e. the main oxidiser valve (MOV), the oxidiser preburner valve (OPOV) and fuel

12

2. State of the Art

Figure 2.3.: Space Shuttle Main Engine schematic [149]

preburner oxidiser valve (FPOV). LH2 and LOX recirculation Wows, through bleed

valves, are maintained for about one hour to chill the four turbopumps down to

cryogenic temperatures and to eliminate gas pockets in the propellant feed system.

Once thermal conditioning is completed, a Vnal dry helium purging of the fuel

lines downstream the MFV is performed [78, 12]. At t = 0 s the start command is

sent and the MFV is opened completely and the three spark igniters are provided

with electrical power. The three main oxidiser valves, MOV, FPOV, and OPOV are

then regulated in order to reach the target priming times for the new preburners

and main combustion chamber (MCC).

These are precisely selected to ensure a stable ignition process and are at 1.4 s

for the fuel preburner (FPB), 1.5 s for the MCC, and 1.6 s for the oxidiser preburner

(OPB). Pressure oscillations in the fuel system, arising form thermodynamic

instabilities during the expander-cycle-like start-up, are closely monitored and

the FPOV and OPOV positions are controlled to avoid high mixture ratios in the

preburners which result in dangerously high temperatures for the turbines.

The FPB is ignited during the second fuel system pressure dip and is followed

by ignition of the PFB. At 1.25 s the rotational speed of the high pressure fuel

13

2. State of the Art

Figure 2.4.: Space Shuttle Main Engine start-up sequence [12]

turbopump is checked to ensure that hydrogen can be bumped through the system

against the back-pressure created by the MCC oxidiser priming. As the drive

power of both high pressure turbines increases, the chamber coolant valve (CCV)

is throttled down to 70%. This conditon is maintained until 2.4 s at which point

the control system measures the MCC pressure and regulates the OPOV, FPOV,

and CCV in order to follow the pre-programmed chamber pressure ramp until

the nominal operational point has been reached. Finally the FPOV is regulated to

adjust the fuel mass Wow rate until the nominaol mixture ratio is obtained. At 5 s

stable operation has been achieved.

During the shut-down phase (see Figure 2.5), the main goal of which is to

ensure a safe and as quick as possible shut-down of the engine, the OPOV is

the Vrst valve to be closed with a closing rate not higher than 45%/s to avoid a

too abrupt thrust decay which would endanger the orbiter’s structural integrity.

Closing of the FPOV follows. Positioning of both valves is monitored to main-

tain a low mixture ratio and maximum oxidiser pressure decay whilst avoiding

14

2. State of the Art

Figure 2.5.: Space Shuttle Main Engine shut-down sequence [12]

back-Wow of hot gases into oxidiser lines. In order to compensate for the increased

heat loads due to throttling, the chamber coolant valve is regulated to force more

coolant into the main combustion chamber. Simulataneously the MOC is closed

at a control rate to ensure a combusiton chamber pressure above the inlet turbine

pressures. Finally after 1 s of additional MFV opening time to assure a very

fuel-rich shut-down, the MFV and the CCV are closed [12, 91].

2.2. Modelling: review of previous works

The development of software tools for analyses of rocket engine systems is critical

to the successful design and Vne tuning of such systems. Typically, the ignition

process of a liquid rocket engine involves non-linear interactions between multiple

engine components with phenomena such as Wow resistance, oU-design turbop-

ump operation, heat transfer, phase change, and combustion. Furthermore, the

physical properties of liquid propellants and combustion products in such systems

vary widely and in a rapid way. Developing tools for predicting the dynamical

15

2. State of the Art

behaviour of an engine with such characteristics is a challenging but important

task for engineers and researchers.

Worldwide, various tools have been developed to simulate the transient behaviour

of rocket engine systems. In 1990, Ruth et al. [124] developed the Liquid Rocket

Transient Code (LRTC) the in-house code of The Aerospace Corporation. It

represents one of the Vrst attempts to simulate propulsion systems with a modular

structure. LRTC models an engine through a modular scheme with the method

of characteristics for a Wow through line segments (pipes) connected by nodes

(zero-dimensional components such as valves, oriVces, pumps, branches etc.).

Comparisons with Titan IV K-01 Wight data of the Stage I start transient demon-

strated general agreement.

The Rocket Engine Transient Simulator (ROCETS) [13] was designed and de-

veloped during the 90s by Pratt & Whitney for NASA-MSFC; it allows for cost-

eUective computer predictions of liquid rocket engine transient performance. The

most popular application of ROCETS is the RL-10A-3-3A rocket engine [13, 14, 15],

varying from start transient analysis to modelling of thrust increase with densiVed

propellants [59, 58].

Another powerful tool created in the United States is the Generalized Fluid

System Simulation Program (GFSSP) for modelling cryogenic Wuids in a complex

Wow circuit [80].

Recently, other researchers and engineers have developed (or have started to)

other codes, for the transient analysis of propulsion systems, but their work was

mainly focused to only a part of an engine system [7, 22, 31].

In Japan, during the 90s, a quasi-steady simulation code for transient analy-

sis of the original LE-7 engine was developed [70]. This was Japan’s Vrst attempt

to develop a staged combustion cycle engine, and establishing a safe and reliable

start-up and shut-down method was very important.

Later in 2002, the Visual Integrated Simulator for Rocket Engine Cycle (VIS-

REC) [6] was developed by Mitsubishi Heavy Industries. VISREC is a one dimen-

sional Wow and heat analysis program using the lumped parameter approach. It

16

2. State of the Art

can analyse start-up and shut-down transient behaviour of many types of engine

cycles such as expander, staged combustion, and gas generator. Together with

the LE-7A rocket engine, the Rocket Engine Dynamic Simulator (REDS) [152] in

2004 was developed and applied to start-up and shut-down transient analyses.

In China, in 2000, Kun et al. [77] have developed a tool to study rocket en-

gine system transient based on the disassembly method whose principles are the

following: the modules have independent physical function and mathematical

model; there are uniform parameters exchange interfaces between each module,

and Vnally, the engine system can be disassembled into modules by practical

physical units.

Also in India, during the 2000s, a Vrst try to develop a “dynamic simulator

for liquid-propellant rocket engines” has been accomplished; this tool is called

CRESP-LP [134].

In Iran, only recently they have transient simulation tools performed by the

University of Technology in Tehran. Prof. Karimi et al. have developed their

own code and performed several analyses to study transient regime in rocket

engines [71, 72, 73, 119]

In Russia, extensive studies have been performed on engine transient behaviour.

These studies have taken advantage of the extensive database generated during

testing of the wide range of liquid rocket engines that Russia has developed. The

problem of transient phase in liquid rocket engine systems has been studied in

[68, 136]. The transient analysis of liquid rocket engine is illustrated in great detail

in [10]. A simple ordinary diUerential equation (ODE) approach is presented and

complemented with experimental-based empirical equations in those cases where

ODEs are not suXcient to describe the occurring phenomena.

Finally in Europe, CNES tested a dedicated library for the modelling and simula-

tion of rocket engine system dynamics developed in the AMESim platform [121].

The platform carries multiple sub-systems of a rocket engine, such as tanks, pneu-

17

2. State of the Art

matic lines, turbopump, regenerative circuit, combustion chamber, and starters.

In cooperation with ONERA, CNES has also developed another tool called

Carins [102, 84], an open platform featuring the “symbolic manipulation” method

to simulate the transient behaviour of propulsion systems.

The Astrium SMART code has been developed for the EPS start-up simula-

tion [74] and recently, from 2008 a complete set of models able to simulate liquid

propulsion system components called European Space Propulsion System Simu-

lation (ESPSS) has been developed by a joint European team in the frame of a

GSTP Programme for the European Space Agency [32].

18

3. ESPSS: European Space Propulsion

System Simulation

This chapter documents the models of the propulsion system library implemented

within the existing analysis software EcosimPro [40], used as a basis for this Ph.D.

research.

EcosimPro is an object-oriented visual simulation tool capable of modelling

various kinds of dynamic systems represented by diUerential-algebraic equations

(DAE) [18] or ordinary-diUerential equations (ODE) and discrete events. The

modelling of physical components is based on the EcosimPro language (EL), an

object-oriented programming language which is very similar to other conventional

programming languages but is powerful enough to model continuous and discrete

processes. It can be used to study both stationary states and transients.

EcosimPro employs a set of libraries containing various types of components

which can be interconnected to model complex dynamic systems:

• control

• math

• mechanical

• ports_lib

• thermal

The European Space Propulsion System Simulation (ESPSS) consists of mul-

tiple libraries to represent a functional propulsion system, e.g. Wuid properties,

pipe networking including multi-phase Wuid Wow, two-phase two Wuids tanks,

non-adiabatic combustion chambers, chemistry, turbomachinery, etc:

19

3. ESPSS: European Space Propulsion System Simulation

• fluid_properties

• fluid_flow_1d

• comb_chambers

• tanks

• turbo_machinery

The Libraries sections hereafter describe those libraries [43], focussing on their

physical modelling, and on the main models used for this Ph.D thesis.

3.1. Fluid Properties Library

fluid_properties is an EcosimPro library in charge of the calculation of Wuid

properties. The functions available on this library are mainly used by the

fluid_flow_1d library for the simulation of Wuid systems. The most impor-

tant features are summarized as follows:

• Most of the Wuids used for rockets applications are available

• Fluids are supported in diUerent categories depending on the type used:

- Perfect gases (transport and heat capacity properties obtained from

CEA polynomials (temperature dependent)). Only used in the combus-

tor/nozzle components

- Perfect gases (transport and heat capacity properties interpolated from

tables (temperature dependent))

- SimpliVed liquids interpolated from tables (temperature dependent)

- Real Wuids interpolated from tables considering either liquid, super-

heated, supercritical or two-phase Wow (temperature and pressure

dependent)

- User-deVned Wuids are available. The properties must be deVned in

external data Vles and can be of any of the last three types previously

mentioned

20


• Mixtures of a real Wuid with a non-condensable gas are allowed. The

homogeneous equilibrium model is used to calculate the properties (quality,

void fraction, etc) in case of two phase Wow. Mixtures of two real Wuids are

not allowed. Therefore, phenomena such as fractional distillation are not

modelled.

The fluid_properties library does not contain any component. It only provides

a large collection of functions returning the value of a Wuid property (or the com-

plete thermodynamic state) by introducing relevant parameters (i.e. temperature,

internal energy, pressure, density, heat transfer and friction correlations etc).

3.1.1. Perfect Gas properties according to CEA

The programming is based on the perfect gas state equation. The expressions used

are summarized as follows:

P = ρR · TMW

Z = 1; β = 1/T ; κ = 1/P

The expressions used for the energy calculation are based on the computation

of the speciVc heat at constant pressure for ideal gases (Cp0) as a function of

temperature only (by means of polynomial expressions). The expression proposed

was obtained from a very large database providing data for a very wide range of

21


temperatures (between 200 and 20.000 K) and is summarized as follows:

Cp

R= a1T

−2 + a2T−1 + a3 + a4T + a5T

2 + a6T3 + a7T

4 (3.1)

H = H(T0) +

∫ T

T0

Cp(T )dT =

R · T ·(−a1T

−2 + a2T−1 lnT + a3 + a4

T

2+ a5

T 2

3+ a6

T 3

4+ a7

T 4

5+b1T

)(3.2)

S = S(T0, P0) +

∫ T

T0

Cp(T )

TdT − R

MWlog(

P

P0) = − R

MW·

log

(P

P0R

(−a1T

−2 − a2T−1 + a3 lnT + a4T + a5

T 2

2+ a6

T 3

3+ a7

T 4

4+b2T

))(3.3)

The functions giving the viscosity and the thermal conductivity in case there

are data available (mainly from NIST species database [88], in this case the NIST

database is not as extensive as for the thermodynamic properties) are also based

on polynomial expressions. The viscosity and thermal conductivity functions have

respectively the following form:

lnµ = A1 · lnT +B1

T+C1

T 2+D1; lnλ = A2 · lnT +

B2

T+C2

T 2+D2

Otherwise the properties are estimated as follows:

Viscosity: There are diUerent estimation methods available. The approach used

is subjected to the availability of property data: critical properties and the dipole

moment. In this case the expression used [120] is:

µ =

[0.807T 0.618

r − 0.357 exp(−0.449Tr) + 0.340 exp(−4.058Tr) + 0.018]· F o

P · F oQ

ζ · 107

where Tr is the reduced temperature computed as follows Tr = T/Tc, and F oQ are

correction factors and ζ is the reduced inverse viscosity, calculated as follows:

ζ = 0.176 ·(

Tc

MW 3 P 4c

)1/6

22


F oP is a correction factor that mainly depends on the polarity of the molecule. It

is computed as follows:

F oP = 1 0 ≤ µr ≤ 0.022

F oP = 1 + 30.55 · (0.292− Zc)

1.72 0.022 ≤ µr ≤ 0.075

F oP = 1 + 30.55 · (0.292− Zc)

1.72 |0.96 + 0.1 · (Tr − 0.7)| 0.075 ≤ µr

F oQ is a correction factor used only in quantum gases. In the present case its

value is 1. µr is the relative dipole moment computed as follows:

µr = 52.46 · µ2Pc

T 2c

where µ is the dipole moment in Debyes.

In case critical properties and dipole moment are not available, other estimations

must be done based on quantum formulation. The following expression will be

used:

µk =ηns ·

√MWk · T

Ωk · 107

where ηns is a constant (26.6958 in S.I.) and Ωk

Ωk = ln

(50 ·MW 4.6

k

T 1.4

)

Thermal Conductivity: Similarly to viscosity, the thermal conductivities that

are not available are estimated. The approach used is summarized as follows:

λk =µk R (3.75 + 1.32 ·

(Cpk/R− 2.5)

)MWk

3.1.2. Perfect Gas interpolated properties

For the state equation, the same expressions as in Chapter 3.1.1 are used. The

expressions used for the energy calculation are based on the table interpolation of

23


the speciVc heat at constant pressure (C0p ) as a function of temperature only:

Cp

R=Cp

R(T )

H = H(T0) +

∫ T

T0

Cp(T )dT

S = S(T0, P0) +

∫ T

T0

Cp(T )

TdT − R

MWlog(P/P0)

The functions giving the viscosity and the thermal conductivity are also inter-

polated from the external user-deVned property Vle as a function of temperature.

3.1.3. SimpliVed Liquid interpolated properties

For simpliVed liquids, the formulation is based on the tables where, density, sound

speed and speciVc heat are interpolated as functions of the temperature. Thus, the

volumetric expansivity can be obtained as follows:

β =−1

ρ

dρ

dT

∣∣∣P=const

This derivative is calculated numerically. Then, the following others thermody-

namic derivatives can be calculated:

Cv =Cp

1 + T · β2 ν2sound/Cp

κ =1

ρ

dρ

dP

∣∣∣T=cte

=Cp

ρ ν2soundCv

Once these properties have been interpolated, the equation of state can be

applied assuming constant compressibility with pressure:

ρ(P, T ) = ρ(T )[1 + κ (P − Pref )

]The enthalpy is calculated integrating numerically the Cp:

H = Hideal(T0) +

∫ T

T0

Cpideal(T )dT

24


Viscosity and Thermal conductivity are also interpolated from the external data

of the property Vle as a function of temperature.

3.1.4. Real Fluids interpolated properties

For real Wuids, FORTRAN functions will perform special searching techniques in

2-D property tables to interpolate the desired property in the nearest cell of the

data tables. A single reading will be done the Vrst time that a function’s call of

any property is made. The functions will identify if a certain Wuid is liquid, vapour

or two-phase Wow by giving a pair of variables that can be ρ-T, ρ-U, s-H , P-H ,

P-T, etc.

With the exception of two-phase Wow or in the case of table extrapolation, no

special hypothesis concerning the Equation of State and the properties has been

done: all the properties are interpolated using the data tables of the properties Vle.

Under two phase conditions the quality of the mixture is calculated from

the saturation properties of the liquid and steam phases. Knowing the mixture

(vapour/liquid) density and energy, the following two equations are used:

quality = x =u− uliq

uvap − uliq

1/ρ = 1/ρliq + x (1/ρvap − 1/ρliq)

An iterative process in pressure is needed. For each iteration, the saturation

conditions will be calculated and the pressure fulVlling the two previous equations

can be found. The void fraction is calculated as follows:

void fraction = α = Vvap/(Vvap + Vliq) = (ρliq − ρ)/(ρliq − ρvap)

The transport properties and the heat capacity in two-phase conditions are

calculated in a simple way:

µ = xµvap + (1− x)µliq (3.4)

λ = xλvap + (1− x)λliq (3.5)

Cp = xCpvap + (1− x)Cpliq (3.6)

25


Nevertheless, the liquid and vapour saturation values of the transport properties,

together with those of the heat capacities, densities and enthalpies (latent heat)

will be returned for the calculation of other two-phase properties as the sound

speed and the Vlm coeXcient.

Under one-phase conditions the sound speed is directly given by the FORTRAN

function interpolations. The returned value is equivalent to the following ex-

pression that uses other properties (Cp, β, κ) also returned by the properties

function:

vsound =

√Cp

ρκCp − β2T(3.7)

Under two-phase conditions the sound speed must be calculated. The equilib-

rium sound speed presents discontinuities at phase changes. In order to ensure

system robustness the following approach is given by Wallis (1969) [147]:

1/v2sound = (αρvap + (1−α)ρliq)(α/ρvap/v

2sound,vap + (1−α)/ρliq/v

2sound,liq) (3.8)

Another “frozen” [28, 8] sound speed expression can be used instead, which is

also continuous with the sound speed at phase changes:

dP

dT=

xCpv + (1− x)Cpl

T (xβvνv + (1− x)βlνl)ν : speciVc volume; x: quality

v2sound =

T

(dP

dT

v)2

x(ε Cpv − T dP

dT νv

((1 + ε) βv − κv

dPdT

))+ (1− x)

(ε Cpl − T dP

dT νl

((1 + ε) βl − κl

dPdT

))where:

ε = 0⇒ “frozen” sound speed

ε = 1⇒ “equilibrium” sound speed

Sub indexes “v” and “l” indicate vapour and liquid saturated conditions.

26


3.1.5. Perfect gas mixtures

Perfect gas mixtures are calculated with linear mixing rules assuming the same

temperature for all the constituents.

xk =ρk

nchem∑k=1

ρk

; Pk = ρkR · TMWk

; P =nchem∑k=1

Pk

where, R is the gas constant = 8314.4 [J/kmol K],MWk is the molecular weight of

the chemical constituent k, T is the mixture temperature [K], ρk is the density of

the chemical constituent k and xk is the mass fraction of the chemical constituent

k.

The molecular weight of the mixture MWmix and the molar fractions yk are

calculated as follows:

1

MWmix=

nchem∑k=1

xk

MWk; yk = molar fraction =

xk MWmix

MWk

The energy properties are computed as follows:

Cp =nchem∑k=1

xk Cpk(T ); Cv =nchem∑k=1

xk Cv,k(T );

H =nchem∑k=1

xk Hk(T ); S =nchem∑k=1

xk Sk(T ) +nchem∑k=1

yk ln yk(T )

The sound speed is calculated as follows:

γ =Cp

Cp −R/MWmix(3.9)

vsound =√γ RT/MWmix (3.10)

Regarding the transport properties, the computation of the mixture viscosity is

27


computed as follows [64]:

µmix =nchem∑

i=1

yi · µi

yi +nchem∑j=1j 6=i

yj · φij

(3.11)

where φij is the interaction parameter estimated with the following formulation:

φij =1

4

[1 +

(µi

µj

)0.5(MWi

MWj

)0.25]2(

2MWj

MWi +MWj

)0.5

Similarly to viscosity, the thermal conductivity for mixtures is computed as

follows:

λmix =nchem∑

i=1

yi · λi

yi +nchem∑j=1j 6=i

yj · ψij

(3.12)

The interaction parameter for the thermal conductivity is based on the one

computed for viscosity. The expression is as follows:

ψij = φij ·[1 +

2.41 · (MWi −MWj) · (MWi − 0.142 ·MWj)

(MWi +MWj)2

]

3.1.6. Real Fluid - Perfect gas mixtures

The Veld of mixture of Wows has been and is still today the subject of intensive

research. Various models exist in the literature that represent with variable

accuracy the chemico-physical phenomena that occur in mixed Wows. There are

mainly two diUerent mathematical formulations of mixed Wows :

• The two-Wuid models, where equations are written for mass, momentum

and energy balances for each Wuid separately.

• The mixture models, where equations for the conservation of physical

properties are written for the two-phase mixture.

Mixture models have a reduced number of balance equations compared to two-

Wuid models, and may hence be considered as simpliVcations in terms of math-

28


ematical and physical complexity. However, some mixture models, like the

Homogeneous Equilibrium Model (HEM) [27] that is used within this work, are

still of signiVcant interest. The HEM formulation has indeed multiple advantages

with respect to the other models :

• its convective part is unconditionally hyperbolic, which is not the case for

other models;

• it is very similar to the Euler equations, thus it can beneVt of all the numeri-

cal studies made for Euler equations;

• we do not need to derive nor to implement the mass, momentum and energy

transfer between phases, as they cancel each other out in the HEM mixture

formulation.

This is why this very simple formulation is still worth being used, despite its

inherent limitations.

The most important restriction is that the Wow should be in equilibrium, or

at least close to it, in order for the HEM formulation to approximate correctly the

physical behaviour of the Wow. By equilibrium one means here that both phases

have the same velocity, pressure and temperature.

The homogeneous equilibrium model of a mixture of two Wuids is then ap-

plied. One Wuid must be real or a simpliVed liquid, and the other a perfect

non-condensable gas (ncg). Assuming that:

• The Real Fluid in possibly subcooled liquid, saturated or superheated vapour

state and the ncg form a homogeneous mixture with a uniform temperature.

• The Real Fluid, if present, occupies the entire volume. ncg, if present,

occupies the same volume as the Real Fluid vapour according to the Gibbs

Dalton Law.

• If ncg and Real Fluid Liquid are present, the Real Fluid vapour is saturated

(relative humidity is equal to 1)

• If ncg is present, the Real Fluid liquid conditions are the subcooled condi-

tions corresponding to P and T. (Liquid phase pressure = Pvap + Pncg),

29


• The ncg gas is insoluble in the Liquid Phase of the Real Fluid. There can be

no ncg if the volume is Vlled with the liquid phase of the Real Fluid

The following state equations are involved in presence of liquid:

ρliq, uliq = fstate(fluid, P, T )

where P = Pnc + Pvap (subcooled conditions)

ρvap, uvap = fsat(fluid, P, T )

where Pvap = fsat(fluid, T ) (saturated conditions)

“u” is the internal energy. Subscript “nc” denotes the non-condensable Wuid. “f”

denote the corresponding pure Wuid functions. In this system of equations, Pnc

and T are unknowns.

Assuming that the volume density, ρ, the non-condensable mass fraction, xnc,

and the mixture energy, u, are known, the following closing equations allow the

calculation of the homogeneous temperature and the non-condensable pressure:

ρnc α = ρ xnc where ρnc = fstate(fluidnc, Pnc, T )

u = (1− xnc)(xuvap + (1− x)uliq) + xnc unc

Applying the deVnitions of the void fraction (α = Vg / Vtot) and quality (x =

Mvap / (Mvap + Mliq)), it is possible to Vnd an expression for the quality appearing

in the equation above as a function of densities:

α = (ρliq − ρcond)/(ρliq − ρvap)

x = α ρvap/(ρliq − α(ρliq − ρvap))

The variable ρcond = (1−xnc)ρ refers to the condensable mass (liquid & vapour)

divided by the total volume. The non-condensable density ρnc and the vapour

density are referring to the gas volume and not to the total volume, so ρnc 6= xncρ.

For the sake of clarity, it can be seen that the last two equations are identities

introducing the deVnition of:

• Vg = Vvap = Vnc

30


• Vtot = Vg + Vliq

• α = Vg / Vtot

• x =Mvap / (Mvap +Mliq)

• ρ = (Mnc +Mliq +Mvap) / Vtot

• ρcond = (Mvap +Mliq) / Vtot

• ρnc =Mnc / Vg

• ρliq =Mliq / Vliq

• ρvap =Mvap / Vg

The heat capacity, viscosity and thermal conductivity are calculated as a mix-

ture of a liquid and a composed gas (the vapour and the non-condensable gas).

The mixture properties are calculated in a simple way (weighing the pure Wuid

properties with the mass fractions):

Cp = xmixCpgas + (1− xmix)Cpliq (3.13)

µ = xmix µgas + (1− xmix)µliq (3.14)

λ = xmix λgas + (1− xmix)λliq (3.15)

where the mixture quality (xmix) is deVned as the mass ratio of gas (vapour + non

condensable)

xmix = α ρgas/(ρliq − α(ρliq − ρgas))

The void fraction α has the same meaning than in a pure two-phase Wuid, i.e.

the gas volume divided by the total Wuid volume. The gas mixture properties are


ρgas = ρvap + ρnc

Cpgas = (ρvapCpvap + ρncCpnc)/ρgas

31


Similarly, the gas mixture transport properties and sound speed are calculated

as follows:

µgas = (ρvap µvap + ρnc µnc)/ρgas

λgas = (ρvap λvap + ρnc λnc)/ρgas

v2sound,gas = ρgas/(ρvap/v

2sound,vap + ρnc/v

2sound,nc)

All individual properties have been computed with pure Wuid functions.

The sound speed is approximated as an equivalent two-phase mixture where

the vapour phase is in fact a mixture of a non-condensable Wuid with 100% of

humidity:

1/v2sound = (αρgas + (1− α)ρliq)(α/ρgas/v

2sound,gas + (1− α)ρliq/v

2sound,liq)

3.2. Fluid Flow 1D Library

fluid_flow_1d is an EcosimPro library for 1-D transient simulations of two-Wuid,

two-phase systems. The most important features are the following:

• The conservation equations include gas, liquid and two-phase Wow regimes

for ideal or real Wuids. The working Wuid(s) can be easily selected from a

large collection of Wuids included in the fluid_properties library.

• The Wuid phase will be automatically calculated. The homogeneous equilib-

rium model is used to calculate a real Wuid under two phase conditions with

or without a non-condensable gas mixture. Absorption/desorption is not yet

considered.

• Flow inversion, inertia, gravity forces and high speed phenomena are con-

sidered in pipes, volumes and junctions, the pipes also incorporating an

area-varying non-uniform mesh 1-D spatial discretisation into n (input data)

volumes.

• Calculation of concentrated (valves) and distributed (pipes) load losses

including two-phase wall friction correlations.

32


• Heat transfer between the walls and the Wuid. Multiple thermo-hydraulic

correlations and initialization options are included.

• Other special components such as check valves, pressure regulators, heat-

exchangers and Tees (for convergent and divergent Wows) are available.

• 1-D Pipe Wows can be simulated using some robust and accurate numerical

techniques upwind (Roe) or centred schemes.

Hydraulic or pneumatic systems where the heat transfer or system controls are

coupled will be easily evaluated with the fluid_flow_1d library. Cavitation and

priming phenomena under two-phase Wow (with or without a non-condensable

gas travelling in a liquid) will be calculated in pipes or other components. Besides,

fluid_flow_1d will permit to analyse in great detail transient aspects due to

inertia (water-hammer) and bubble collapse (priming).

3.2.1. Components ClassiVcation

The components of the fluid_flow_1d Library are listed in Figure 3.1 below,

where the inheritance hierarchy is shown. In an ESPSS Wuid network, every

component is either a resistive component or a capacitive component. A resistive

component receives the state variables (pressure, density, velocity, chemical com-

position and enthalpy) as input and gives back the Wow variables (volumetric, mass

and enthalpy Wows) as output. A capacitive component receives the Wow variables

as input and gives back the state variables at output. To build a Wuid network, the

user must connect resistive components to capacitive ones, alternatively. So, from

a computational point of view, components are divided into two classes:

• C (capacitive) elements, integrating the mass and the energy conservation

equations. The thermodynamic functions will be used to calculate the

complete thermodynamic state

• M (momentum) elements, calculating explicitly (inertia terms) the mass

Wows between C elements. Reverse Wow is allowed

This computational scheme prevents the appearance of algebraic loops and high

index DAE (DiUerential Algebraic Equations) in the mathematical model of the

pipe network.

33


Figure 3.1.: Components in the fluid_flow_1d library

The existing components of the fluid_flow_1d library have the following

types:

• Volumes and Heat-exchangers components are C elements.

• Junctions, Valves, Filter and Jun_TMD components are M elements.

• Bound components (VolPT_TMD, VolPx_TMD, etc) are also C elements.

(Here, TMD means time dependant). VolPsTsVs_TMD and VolPsTs_TMD

are M elements because they calculate mass Wow. In TMD elements, state

variables are imposed in the experiment Vle (Vxed or depending on the time)

or by means of control library components connected to its control ports.

• Pipe, Tube, HeatExchanger and Nozzle components are 1-D models with

dedicated numerical schemes comparable to a C element.

• Pipe_res and ColdThruster components are 1-D models with dedicated

numerical schemes comparable to an M element. The ColdThruster incor-

porates an internal valve component.

• Others topological components such as the Tee component are M elements

because they internally Vnish in junctions, even if they have some internal

34


C components.

The graphical symbols of the components provide information about the kind

of computational element to which each port is connected. Ports belonging to a C

element have a small dot in the middle of the arrow while ports belonging to an

M element are just represented by the arrow (see Figure 3.1). It is noted that the

Tube and Pipe components can simulate an area-varying non-uniform 1-D mesh,

as it is the case for the ColdThruster and Nozzle components, even though, for

simplicity, the symbol graphical representation does not indicate this capability

on Pipes and Tubes components.

3.2.2. Junction/Valve

This component represents a junction. It is a basic component where no mass

accumulation is considered. The mass and enthalpy Wows of the inlet and outlet

ports are equal (no mass accumulation in junctions):

m1 = m2 = m

m1,nc = m2,nc = mnc

mh1 = mh2 = mh

where indexes 1 and 2 refer to the connected Wuid ports. For each port, “m”,

“mnc”, and “mh” are the total mass, non-condensable mass and the enthalpy Wows

respectively. These Wows are calculated here taking into account the Wow direction,

so diUerent temperatures and densities may exist at both sides of a junction.

The following momentum balance equation dynamically calculates the mass

Wow per unit of area:

(I1+I2)

(AdG

dt+G

dA

dt

)+lv

dG

dt= (P+0.5ρv2)1−(P+0.5ρv2)2−0.5(ζ+ζcrit)

G|G|ρup

(3.16)

where, P1 , P2 are the static pressures at port 1 and 2, calculated by the connected

volumes (0.5ρv2)1−2 represent the dynamic pressures at port 1 and 2, calculated

by the connected volumes. ρ and v are the mean density and speed at the

connected volumes; I1 , I2 represent half inertia of the connected pipe ends 1 and

35


2, respectively. A is the instantaneous valve cross section; lv = sqrt(Aref ), this

term (valve inertia) is very small but makes the equation no singular if A = 0.

G is the mass Wow per unit of area, ζ is pressure drop coeXcient and ρup is the

upstream gas/liquid mixture density.

The mass Wow will be calculated as m = GA. The use of G (mass Wow per unit

or area) instead of m (mass Wow) allows a complete closing (A = 0) of the valve

without making the system of equations singular.

The pressure drop contribution is quadratic with mass Wow: 0.5(ζ+ζcrit)G|G|/ρup.

This term would make the momentum equation singular at zero Wow because

very small perturbations in pressure lead to non-negligible variations in mass Wow

(∂G/∂P →∞). Physically, what is happening is that pressures losses are linear

with mass Wow for laminar regimes. To account for that, the quadratic term G‖G‖is linearised for G < Glam in this way:

G|G| =

k(G)G G < Glam

G2sign(G) G > Glam

Glam = µRelam

√A; k(G)G→0 → Glam

k(G) is a smoothing factor to assure continuous transitions between laminar

to turbulent regimes. µ is the upstream viscosity calculated by the connected

volumes, Relam is the minimum Reynolds number set at 2000 as default value.

The sonic Wow limitation is taken into account by adding a correction factor

to the pressure loss coeXcient, ζcrit, which limits the mass Wow per unit area to

be less than or equal (≤) to the critical Wow per unit area. First, the Wow under

steady state conditions is obtained by cancelling the derivatives in the previous

momentum equation:

Gst =√

2ρup [(P + 0.5ρv2)1 − (P + 0.5ρv2)2]

The added term is calculated in such a way that if the Wow attempts to be

greater than critical Wow, the following extra term will limit the Wow to the critical

value:

36


ζcrit = max((Gst/Gcrit)

2 − ζ, 0)

where “Gcrit” is the critical (sonic) Wow per unit of area (ρc)crit calculated by

the capacity components (see section 3.2.3) connected by the junction.

3.2.3. Capacity/Volume

The Capacity component simulates a volume with several Wuid ports named f[j].

It’s the basic capacitive component containing the mass and energy conservation

equations for this type of components.

Here below the general equations for a non-adiabatic variable volume. It is

assumed that the mixture (non-condensable plus main Wuid in liquid, gas or two

phase conditions) has an homogeneous temperature.

Mass conservation

Vdρ

dt+ ρ

dV

dt=

∑j∈Ports

mj (3.17)

Non-condensable mass fraction xnc conservation

ρ Vdxnc

dt+ xnc

(Vdρ

dt+ ρ

dV

dt

)=

∑j∈Ports

mncj (3.18)

Energy conservation

Vdρ

dtu+ ρ

dV

dtu+ ρV

du

dt=

∑j∈Ports

(mh)j + Q− PdV (3.19)

u = total speciVc energy = ust + v2/2 (3.20)

where ρ, xnc and u are the Wuid mixture (including two phase Wow) density, the

non-condensable mass fraction and the total energy respectively; mj , mncj and

mhj are the mass and enthalpy Wows at port j calculated at the connected resistive

type components (see section 3.2.2). V is the volume, which can change with time.

Assuming that the volume V and its rate of change are known, previous

conservation equations enable to calculate the derivatives of the mixture density,

37


mixture energy and non-condensable mass fraction. These variables can be

integrated, so they are known at any time.

Assuming thermodynamic equilibrium, the conservation equations are always

valid even if the Wuid conditions are liquid, vapour or homogeneous two phase

Wow. Then, the complete thermodynamic state (partial pressures, temperature,

quality ...) can be calculated using the pure Wuid thermodynamic routines or the

homogeneous equilibrium model for mixtures of a non-condensable gas plus a

real Wuid: FL_state_vs_ru

Volume average velocities are required for the total energy conservation equa-

tion, the evaluation of the wall frictional forces and of the wall heat transfer. For

the calculation of the average velocity, volumes are considered to have two sides,

side 1 and side 2. The total mass Wow rate entering the volume at side 1 and 2 is:

m1,in =∑

∀ports in side1

mj,in; m2,in =∑

∀ports in side2

mj,in

Port mass Wows can be positive or negative. It is deVned as positive when

entering the volume. The average velocity in the volume is deVned as:

v =min,1 − min,2

2ρA

where ρ is the average density in the volume, and A is the cross area of the

volume. This average volume velocity is transmitted to the ports. The eUective

port velocity will be multiplied by the cosine of the port angle α because the

lateral velocities do not compute in the total pressure:

v(j) = v cos(αj)

The term Q appearing in the energy conservation equation of the capacity

permits the exchange of heat through a thermal port. The walls (that can be

represented by thermal components) are not included in this component:

Q = hfilmAwall (tp.T − T )

where tp is the name of the thermal port (with one node) connected to the

38


Volume. “tp.T” behaves as the internal wall temperature, to be determined in the

connected thermal component. The Vlm coeXcient is calculated using empirical

correlations.

3.2.4. Tubes/Pipes

These components simulate area-varying non-uniform mesh high resolved 1D

Wuid veins that exchanges heat with a 1D thermal port. They incorporate the

1D mass, energy and momentum equations in transient regime. The number of

volumes in which the pipe is discretised will be a parameter.

All kind of Wows (compressible or nearly incompressible Wows, single component

or two-component Wows, single phase or two-phase Wows) can be simulated by us-

ing the following system of governing equations, here in area-scaled conservation

form [138]:

Mass conservation:

A∂ρ

∂t+∂ρvA

∂x= −ρAkwall

∂P

∂t(3.21)

Non-condensable mass fraction xnc conservation:

A∂ρxnc

∂t+∂ρvxncA

∂x= −ρxncAkwall

∂P

∂t(3.22)

Momentum conservation:

A∂ρv

∂t+∂[(ρv2 + P )A]

∂x= −1

2

dξ

dxρ v|v|A+ ρgA+ P

(dA

dx

)(3.23)

Energy conservation:

A∂ρE

∂t+∂ρvHA

∂x=

(dQw

dx

)+ ρgvA (3.24)

where ρ, xnc, P , u are the gas/liquid mixture density, the non-condensable mass

fraction, the pressure and the total energy respectively. A is the variable Wow

area and v the velocity. This system of 4 equations represents the general case

39


of a mixture of two Wuid components, for which the Vrst one can be either one

phase or two-phase, and the second one is always a non-condensable gas. This

set of equations is closed by a thermodynamic equation of state (EoS), which is

described in the Wuid properties library, and hereafter written under general form:

p = p(ρ, u) (3.25)

The choice of density ρ and internal energy u as independent thermodynamic

variables is the most eXcient one regarding CPU-time when the EoS is left under

arbitrary form.

The diUerent source terms are the following:

• In the Vrst equation governing the mixture mass conservation, a source term

responsible for the wall compressibility eUect of the mixture, determinant

in water hammer simulations, is included; kwall is the wall compressibil-

ity. Assuming linear elasticity for the pipe wall material, we have three

conVgurations:

pipe anchored with expansion joints throughout: kwall = Din/t/ME

pipe anchored at its upstream end only: kwall =Din

tME

(5

4− η)

pipe fully anchored: kwall =Din

tME(1− η2)

ME is the Young’s modulus of elasticity, η is the Poisson’s ratio and t is the

wall thickness.

The wall compressibility shall be multiplied by ∂P/∂t to account for the

volume change. For this purpose is calculated from the current state variables

(density and energy) and the thermodynamic derivatives:

∂P

∂t=

(∂ρ

∂t− ∂ρ

∂h

∣∣∣P

(∂u

∂t+P

ρ2

∂ρ

∂t

))/

(∂ρ

∂P

∣∣∣h− 1

ρ

∂ρ

∂h

∣∣∣P

)

• In the second governing equation, the non-condensable mass conservation

(if any), a similar source term is included;

• In the third governing equation, the mixture momentum conservation, a

source term represents the friction (dξ, proportional to dx, is the pressure

40


drop coeXcient derived by empirical correlations), another one takes into

account the gravity, and the last one is responsible for the area variation.

The equivalent distributed friction, ∆ξi is calculated as follows:

∆ξi = kadd +∑

bend,j

hdcbend(αbend,j , Rbend,j , Di, ε)+∆xi

Dihdcfric(Di, ε, Rei)

where kadd is an input data representing concentrated load losses to be

distributed along the pipe; Function hdcbend calculates the bend pressure

drop coeXcient; Function hdcfric calculates the friction factor including

laminar and turbulent regimes.

“g” represents the gravitational acceleration, if any. It is computed as the

scalar product of the gravity vector (gx, gy, gz) with the direction of the pipe

in the global axis system (∆x, ∆y, ∆z), which are the diUerence of position

of the tube tips.

• In the last governing equation, the mixture energy conservation, a source

term Qw takes into account the heat transfer with the wall when it is

included:

Qw = hfilm dx[Pinner(tp_in.T − T ) + Pouter(tp_out.T − T )]

where Pinner and Pouter are the wet perimeters; tp_in, tp_out are the names

of the thermal ports connected to the tube with the same number of nodes

as the Wuid vein.

The port temperatures tp_... T behave as the wall internal temperatures, to

be determined in the connected thermal components. The Vlm coeXcient

for each node is calculated using empirical two-phase correlations:

hfilm = htc(x, Dh, λ, Re, Pr . . .)

Re =GDh

µ

Pr =µCp

λ

41


where G = ρv is the mass Wow rate per unit area, Dh is the hydraulic

diameter of the pipe Dh = 4A/Pw. In case of circular cross section Dh is

equal to the geometric diameter of the cross section; otherwise it represents

a reasonable characteristic length of the cross section.

Another source term, ρgvA, takes into account the gravity work.

The tube and pipe components are discretised by either a centred or an upwind

numerical scheme. Figure 3.2 describes the pipe discretisation. The inner Wuxes are

computed using one or the other of these schemes, and the Vrst and last junctions

(1 and n + 1) ones are given by the Wuid ports, as they are calculated at resistive

type components using momentum equation with sonic Wow limitation. Note

that the Vrst and last half-nodal inertia are included in the junction component

equations.

Figure 3.2.: Pipe discretisation

Using the centred scheme, a staggered mesh approach is applied, for which the

state variables (pressure, density, velocity, chemical composition and enthalpy) are

associated with the n nodes, and the Wow variables (volumetric, mass and enthalpy

Wows) are calculated at the internal junctions (each junction has associated two

half volume inertias). With this scheme, the various Wuxes to be computed at the

inner junctions are simply the Wow variables, except for the mixture momentum

42


Wux that is associated to the n nodes:

fmass,j = mj

fnc,j = mncj (3.26)

fmom,j = (Pi + ρiv2i + qni)Ai

fene,j = mHj

The momentum Wux term includes an artiVcial dissipation term qni calculated as

follows:

qni = −Dampmi+1 − mi

Avsound,i

As an alternative to the centred scheme, an adequate upwind scheme has been

developed to improve the discontinuities resolution of these transient two-phase

Wows in quasi-1D pipe networks. Using that scheme, a collocated mesh approach

is applied, for which all variables are discretised on the n nodes, even the mass

Wow. All the Wuxes f(u) are discretised at the junctions and include a central

part and an upwind part, following the initial Roe scheme [123]. Please refer to

Appendix A for a detailed description of the scheme.

3.3. Turbomachinery Library

turbo_machinery is an EcosimPro library for the simulation of pumps, turbines

and compressors. The most important features are the following:

• Pump model provided with user-deVned dimensionless turbo-pump charac-

teristic curves adapted to positive and negative speeds and Wow zones

• Turbine and Compressor components provided with user-deVned dimension-

less performance maps as a function of the reduced axial speed and pressure

ratio

• Special turbomachinery components allowing simple calculation of general-

ized performance maps as a function of the nominal performances and other

signiVcant design data

43


• Programming of the turbomachinery components is non-dependant on the

working Wuid type: the properties of the selected working Wuid (calculated

inside the corresponding thermodynamic function) will depend on the Wuid,

but the non-dimensional parameters of the performance maps are equally

deVned for all kind of Wuids.

The generalized performances maps of the turbo_machinery library allows

robustly analysing the transients during the start up and shutting down processes,

where the reduced axial speed and Wow are far away from the nominal values. The

turbo_machinery components can be connected to fluid_flow_1d components

with the aim of simulate a complete rocket engine cycle.

The components of the turbo_machinery library are listed below, and repre-

sented in Figure 3.3:

- Compressor

- Pump

- Turbine

- Compressor generic

- Pump generic

- Turbine generic

- Pump vacuum

Two diUerent types of pumps/turbines are available: one “generic” model if the

oU-design characteristics are unknown and one speciVc model which can only be

used with tables, for well deVned turbo machinery.

All these components behave externally as resistive elements (see section 3.2.1).

Components ports are resistive because they calculate the mass Wow. Nevertheless,

every model of a turbo-machinery component includes an internal capacitive

element receiving/giving the mechanical work.

Since in turbopump-fed engine cycles only pumps and turbines are present, only

these two class of components will be discussed in this section.

44


Figure 3.3.: Components in the turbo_machinery library

3.3.1. Pump & Generic Pump

This component simulates a pump for liquids. It is provided with Vxed or user-

deVned dimensionless turbo-pump characteristic curves adapted to positive and

negative speeds and Wow zones, and valid for several types of pumps.

Usually it is rather diXcult to Vnd pump curves including the non normal zones,

so this is the reason to include in this component a set of curves [25] covering all

zones of the pump operation for 3 diUerent speciVc speeds: Ns = 25 corresponding

to a pure centrifugal pump, Ns=147 corresponding to mixed pump, and Ns=261

corresponding to an axial pump. The speciVc speed is deVned as:

Ns =rpm

√Q/ns

(TDH/nst)0.75

where, ns is number of suctions, nst is the number of stages, Q [m3/s] is volumetric

Wow and TDH [m] is the actual total dynamic head of the pump.

The pump model makes use of performance maps for head and resistive torque.

The pump curves are introduced by means of Vxed 1-D data tables deVned as

functions of a dimensionless variable θ that preserves homologous relationships

45


in all zones of operation. θ parameter is deVned as follows:

θ = π + arctan(ν/n) (3.27)

where ν and n are the reduced Wow and speed parameters respectively:

ν =Q

QR=min/ρin

QRn =

ω

ωR(3.28)

The dimensionless characteristics (head and torque) are deVned as follows:

h =TDH /TDHR

n2 + ν2β =

τ / τRn2 + ν2

(3.29)

τ and TDH are the torque and the total dynamic head respectively. Sub index R

means “rated” (nominal) conditions. The nominal torque is calculated from the

other nominal parameters:

τR =g ρup TDHRQR

ηR ωR(3.30)

This method eliminates most concerns of zero quantities producing singularities.

To simplify the comparison with generic map curves, these relations are normal-

ized using the head, torque, speed and volumetric Wow at the point of maximum

pump eXciency.

In case there are no user deVned curves, the ones already implemented in the

component at diUerent speciVc speeds will be used and interpolated as function of

the actual Ns and θ:

h = h_vs_theta_Ns(Ns, θ) dimensionless pump TDH

β = β_vs_theta_Ns(Ns, θ) dimensionless pump torque

using the deVnition of h and β, the actual torque τ and the pressure rise TDH

(expressed as the total dynamic head in meters) will be calculated.

The mechanical balance allows the calculation of the axial speed dynamically:

Imechdω

dt= τshaft − τ (3.31)

46


where ω is mechanical speed, Imech is the mechanical inertia and τ is the torque

calculated using the non-dimensional performances pump.

The enthalpy Wow rise is a function of the absorbed power while the evaluation

of the mass Wow rate is performed through an ODE.(m h)out = τ · ω − (m h)in

I · dmdt

=(P + 1

2ρv2)out−(P + 1

2ρv2)in− gρin · TDH

(3.32)

In case performance maps of a particular pump are known, a diUerent approach

is used; the pump curves (head and torque) are introduced by means of input data

tables:

Independent variables:

- Mass Wow coeXcient: φ+ = m/(ρin ω)

Dependent variables:

- Head rise coeXcient: ψ+ = ∆P/(ρin ω2)

- Reduced torque: C+ = τ/(ρin ω2)

The head rise and needed torque coeXcients are computed with 1-D tables,

as a function of mass Wow coeXcient only. Rotational speed is not taken into

account. The pump model described here has the disadvantage of being less

general than the Pump_gen component: the coeXcients used (φ+, ψ+ and C+)

are not dimensionless. Hence, the characteristics diUer for each pump, even

for geometrically similar pumps (with impellers having the same angles and

proportions).

3.3.2. Turbine & Generic Turbine

This component simulates a turbine for gases. It is provided with calculated (no

input) but adjustable dimensionless characteristic curves adapted to positive and

negative speeds and Wow zones, and valid for several types of turbines. Adjustable

dimensionless pressure ratio curves are function of beta design parameter, reduced

47


speed and inlet Mach number. EXciency curves are function of reduced speed

and pressure ratio parameters.

The value of the power W is obtained using GasTurbo_pow function. This

function basically estimates the eXciency as a function of the reduced speed and

pressure ratio parameters. It also needs the inlet mass Wow to calculate the power.

The rest of the arguments are input data (ηnom, Nnom,...) or calculated values in

the connected pipes components (bound pressures and enthalpies, etc.).

As for the pump, the mechanical balance allows the calculation of the axial

speed dynamically:

Imechdω

dt= τshaft − τ

The mechanical work (τ · ω) extracted from the turbine is simulated as an

enthalpy Wow:

(m h)out = (m h)in − τ · ω

The inlet mass Wow equation is expressed dynamically in accordance with the

turbine pressure drop as follows:

I · dmdt

=

(P +

1

2ρv2

)in

−(P +

1

2ρv2

)out

− (Πnom − 1)Pin · dp_rel

dp_rel is obtained using GasTurbo_dpTurb function. This function basically

estimates the relative pressure drop of the turbine as a function of the beta design

parameter, the reduced speed and the inlet Mach number.

The inlet Mach numberMin is related to the inlet mass Wow: Min =min

ρinAinCoCo is the sound speed calculated at the outlet volume or at inlet according to the

vsound_outlet option.

When speciVc turbine performance maps are introduced, the Turbine_Gen

component cannot be used anymore and the Turbine component is then necessary

adopted. Turbine map curves (dimensionless torque and mass Wow coeXcients)

are input data tables depending on the dimensionless speed and pressure ratio

coeXcients.

The performance maps (mass Wow coeXcient and speciVc torque) are introduced

by means of 2D input data tables:

Independent variables:

48


- Speed coeXcient: N =r · ωCo

- Total pressure ratio: Π = P01 /P02

Dependent variables:

- Mass Wow coeXcient: Q+ =mmap · Co

r2 P01

- SpeciVc torque: ST =τ

r mmapCo

Independent variables are easily calculated because they are obtained from

the dynamic rotational speed and from the boundary pressures. Then, the mass

Wow and torque will be computed interpolating into the 2D input data tables

representing the turbine maps. Compared to the generic component model, only

the mass Wow equation is diUerent. The inlet mass Wow equation is expressed

dynamically in accordance with a time delay (inertia terms):

τ · dmdt

= (mmap − m) ; τ = I · r2/Co

3.4. Combustion Chambers Library

comb_chambers is an EcosimPro library for the simulation of rocket engines. The

most important features are the following:

• The properties of the combustion gases (transport and heat capacity) are

obtained from the CEA [55] coeXcients (see Section 3.1.1) for an arbitrary

mixture or chemical’s reactants. The equilibrium molar fractions at of a

mixture of reactants are derived from the Minimum Gibbs energy method

• Non adiabatic 1D Combustor component: the equilibrium combustion gases

are calculated using previous capabilities. The chamber conditions will be

derived from the general transient conservation equations along a 1D spatial

discretisation

• Inclusion of another more advanced non-equilibrium, non adiabatic 1D

combustor component where a model for the liquid droplets evaporation

and for the global reaction time are also considered. The non-equilibrium

49


model is only a Vrst approach; it does not include Vnite rate chemistry but

only time delay parameters

• Pre-burners and main combustion chambers are topologically built by means

of a combustor, two injectors and two cavity components. They can work

either under liquid, two-phase or gas injection conditions. The bubble

collapse calculation is included in the model of cavities

• The combustion gases generated in a chamber can be conducted (using

standard fluid_flow_1d components) to the turbines or even to other

chambers where any of the previously combusted gases will be considered a

new reactant.

• Cooling jacket component: Several models are available: one with a com-

plete 3D wall temperature distribution and another also including the injec-

tion tores

• Modelling of solid propellant starters, igniters and thermal coating protection

are available using combustor components

• Ideal or non adiabatic exhaust nozzles provided with a 1D spatial discreti-

sation. A special nozzle component allows simple simulation of the Vlm

cooling injection

The comb_chambers components can be connected with fluid_flow_1d,

tanks or turbo_machinery components for the simulation of a complete rocket

engine cycle. Models where one or more chambers are present (staged engines)

can be evaluated. This library permits to analyse in great detail the transients

during the start-up and shut-down processes, where the valve sequences are

decisive.

The numeric method used for the resolution of the subsonic sections of a

combustor is based on the transient conservation equations. ESPSS combustor

models calculate the chamber pressure and the mixture ratio as a function of the

combustor geometry (design parameters) and the physical boundaries whereas in

CEA code those variables were imposed. Advantages of the ESPSS methods are:

50


• Transient phenomena (including pressure/temperature peaks at the start up

and shut down processes) are taken in to account

• The number of implicit equations is reduced, the state variables being

dynamic

• The wall heat exchange (non adiabatic terms) and the pressure drops are

taken into account

• Transient formulation allows to include vaporization / non-equilibrium

phenomena, as it done in the Combustor_rate component

The drawbacks of this method are:

• The characteristic time (integration time step) can be very small. Neverthe-

less, numerical instabilities are normally smoothed, the integration being

faster than using an implicit method

• The total pressure is not strictly conserved along the 1D combustor volumes.

Typical errors are between 0.5 and 1%, mainly produced near the throat

where the Mach number is close to 1

Concerning the supersonic sections of nozzles, a resolution method based on

the transient conservation equations would have numeric problems (passage from

subsonic to supersonic regime, shock waves if non-adapted conditions, etc), so

a 1D quasi-steady implicit method has been implemented for these components,

including non-isentropic eUects under frozen or equilibrium conditions.

The components of the comb_chambers library are listed in Figure 3.4, where

the inheritance hierarchy is shown. The existing components have the following

types (see section 3.2.1):

• Cavities, Regenerative Circuit and Nozzle components are C (capacitive)

elements

• Injectors and Chemical inWator components are M (resistive) elements

• Propellant Wuid ports of Preburners and CombustChamberNozzle compo-

nents are capacitive, and can be connected to any fluid_flow_1d library

51


resistive component. Thermal ports should be connected to a Cooling Jacket

component or other thermal component

Figure 3.4.: Components in the comb_chambers library

ABS_Combustor, Injector and Inj_Cavity components should only be used for

building Combustion Chambers and Preburners because they must be intercon-

nected to each other.

Preburner components include a combustor, two injectors and cavities compo-

nents, and an outlet resistive Wuid port where the mass Wow is calculated. This port

must be connected to a fluid_flow_1d capacitive component so the combustion

gases can be conducted to another combustor.

CombustChamberNozzle components include a combustor, two injectors and

cavities components, and a non-adiabatic 1D Nozzle. They have a special exit port

so a new nozzle extension component (with or without Vlm cooling injection) can

be connected.

52


Th_mux, Th_demux are multiplexer/demultiplexer components allowing split-

ting or jointing a vectorized thermal port into several ones, so that diUerent sized

Cooling Jacket components can be connected to a chamber.

3.4.1. Injector Cavity

Inherited from a Capacity (fluid_flow_1d library, see Section 3.2.1) this compo-

nent represents the combustion cavities upstream the injectors with thermal ports

allowing heat exchange.

The formulation of the conservation equations are the same as in Section 3.2.3

they are valid even if the Wuid conditions are liquid, vapour or homogeneous two

phase Wow.

This component is also charged of the calculation of the propellant molar

fractions from the injected Wuids:

Nchem = 1; (case of pure Wuid)

Nk = yk; MWmix =Nchem∑

k=1

yk MWk; (Case of previously burned gases)

In case of a pure Wuid, chem is the chemical corresponding to the main Wuid. In

case of previously combusted gases, yk are the molar fractions at the inlet of the

cavity calculated by an upstream combustor. MWk is the molecular weight of the

chemical constituent k.

3.4.2. Combustor Equilibrium

In the combustion chamber the merged, mixed and atomized propellants are va-

porized and burned. In doing so, the chemical bound energy from the propellants

is transformed into thermal energy. Hence, it follows an increase of the combus-

tion chamber temperature Tc, which also involves a pressure increase Pc in the

chamber.

This component represents a non adiabatic 1D combustion process inside a

chamber for liquid or gas propellants. The transient conditions (pressures, tem-

peratures, mass Wows and heat exchanged with the walls) will be derived from

general 1D transient conservation equations.

53


The equilibrium combustion gases are calculated using the minimum Gibbs

energy method as a function of the propellant’s mixture molar fractions and

enthalpies, and the chamber pressure.

A mixture equation between the injected propellants and the combustion gases

is applied. From the deVnition of the mixture ratio (MR) and derivation, the

following dynamic equation gives theMR evolution:

MR = Massox/Massfu

Massfu =(ρV )chamber

1 +MR

⇒ mox = MR · mfu +dMR

dt

(ρV )chamber

1 +MR(3.33)

The implementation of the 1D set of equations is done according to a staggered

grid in which the P /ρ/x variables are deVned in the centre of the volumes and the

mass Wows at the junction between the volumes.

Gas mixture mass conservation equation:

A∂ρ

∂t+∂ρvA

∂x= 0 (3.34)

Gas mixture momentum conservation equations:

A∂ρv

∂t+∂[(ρv2 + P )A]

∂x= −1

2

dξ

dxρ v|v|A+ P

(dA

dx

)(3.35)

Gas mixture energy conservation equation:

A∂ρE

∂t+∂ρvHA

∂x=

(dqwdx

)(3.36)

At injector level (i=0), the junction mass Wows will be calculated by the Injector

components. The eUective liquid mass Wow entering into the chamber will be

computed as follows:

• Burning = FALSE, only the injected vapours and non-condensable gases

contribute to the chamber pressurisation. Injected liquids are supposed to be

lost

54


• Burning = TRUE, it is supposed that all the injected liquid will be vaporised

within a delay time, τv (injected gas is not modiVed):

dmfu,liq

dt=

((1− xfu)mfu tanh

(10Tn − Ttr,fu

Ttr,fu

)− mfu

)/τv

dmox,liq

dt=

((1− xox)mox tanh

(10Tn − Ttr,ox

Ttr,ox

)− mox

)/τv

where mox,liq, mfu,liq are the eUective injected mass Wow; mox, mfu are

the total injectors’ mass Wows; xfu and xox are the gas (vapours and non-

condensable gases) mass fractions in the cavities. The hyperbolic tangent

term is added to produce a relaxation of the injected liquid mass Wow at

very low temperatures (ignition process)

Total enthalpies hjun are calculated using the upstream cell conditions: hjun,i =

hi−1 = (u+ P/ρ)i−1.

Again, at injector level the enthalpy Wows will be computed using the eUective

liquid and gas mass Wows multiplied by the corresponding cavity enthalpy (liquid

and vapour). In this way injection conditions (liquid, gas or two-phase Wow) will

be taken into account.

The starter terms (starter_m and starter_mh) will be added to the mass and

energy conservation equations of the Vrst chamber volume. The composition of

the solid propellant gases is an input data within a predeVned set of chemicals

starter_mh = f(starter_T, powder_composition);

starter_m, starter_T and powder_composition being input data.

In the momentum Wuxes an artiVcial dissipation qn is added, and is calculated

as follows:

qni = −Dampmjun(i+ 1)− mjun(i)

Avsound(i)

Damp is a global input data of the fluid_flow_1d library. Momentum equa-

tions are applied to the exit of any volume in which the combustor is discretised

with the exception of the last volume that should end with the throat to avoid

numerical problems (transition from subsonic to supersonic Wow). The outlet mass

Wow will be calculated at the throat.

Node velocities v are calculated using the adjacent junction mass Wows and the

55


mean junction densities:

vi =1

Ai

(mjun,i−1

ρi + ρi−1+

mjun,i

ρi + ρi+1

)i = 2, n; v1 =

mjun,1

A1 ρ1(3.37)

For the Vrst volume, it is supposed that only the outlet junction will account for

the node velocity. For the last volume, the throat density is used.

The vapours and the non-condensable conservation equations take into account

the mixture process: (∂xfuρ

∂t

)A+

∂xfumjun

∂x= 0(

∂xoxρ

∂t

)A+

∂xoxmjun

∂x= 0(

∂xncρ

∂t

)A+

∂xncmjun

∂x= 0(

∂xpwρ

∂t

)A+

∂xpwmjun

∂x= 0

where, xfu is the reducer vapour mass fraction, and xox is the oxidizer vapour

mass fraction ; xnc is the non-condensable mass fraction and xpw represents the

solid propellant gases mass fraction.

The mass Wow of the injected vapours (mfu,mox) and injected non-condensable

gases will be added as source terms to the respective conservation equation of

the Vrst chamber volume. In the same way, the mass Wow of the solid propellant

gases, starter_m, will be added as source a term to the solid propellant gases

conservation equation of the Vrst chamber volume.

Under burning conditions, only the mass fractions of the Vrst chamber volume

will be used to compute the molar fraction of the reactants. Subsequent volumes

will use as reactant the product of the upwind volume.

For each chamber volume the combustion gases properties (product’s molar

fraction, heat and transport properties) are calculated using the Minimum Gibbs

energy method as a function of the propellant molar fractions, pressure and

speciVc enthalpy:

56


Reducer contribution:

Nk,i = xfu,iyk,fu

MWmix,fu; MWmix,fu =

Nchem∑k=1

yk,fuMWk,fu

Oxidiser contribution:

Nk,i = Nk,i + xox,iyk,ox

MWmix,ox; MWmix,ox =

Nchem∑k=1

yk,oxMWk,ox

Solid propellant gases contribution:

Nk,1 = Nk,1 + xpw,iyk,pw

MWmix,pw; MWmix,pw =

Nchem∑k=1

yk,pw MWk,pw

Non-condensable gases contribution:

Nnc,i = Nnc,i +xnc

i

MWnc

Nchem is extended to any chemical treated by the fluid_properties library;

MWk is the molecular weight of the chemical constituent k; xnc1 is the mass

fraction of non-condensable gas at volume 1.

xfu,i, xox,i, xpw,i are mass fractions of vapours and solid propellant gases;

yk,fu; yk,ox; yk,pw are the molar fraction of chemical k of the reducer (oxidizer)

mixture and Nk,1 is number of moles of the chemical constituent k of the reactant

mixture at volume 1.

The propellants (reducer and oxidizer mixtures) molar fractions have been

calculated by the cavity components. Propellants can be formed by any allowed

Wuid mixture using the fluid_flow_1d library, including those of a previous

combustion.

Then, the number of moles, Nk is normalized. With this entry calculated and

using the “dynamic” enthalpy value obtained from the conservation equations of

the Vrst combustor volume it is possible to call the Minimum Gibbs energy method

to obtain the equilibrium temperature and the molar fraction of the products:

(yk_eq, Teq) = fminGibbs(Nk,1, h1 − v21/2, P1)

57


Two possibilities are foreseen calling previous function:

- Equilibrium

- Frozen Wow

In the last case (no ignition), molar fractions remain constant: yk_eq,1 = Nk,1.

Under frozen conditions, the molar fractions of the subsequent volumes will be

calculated as for the Vrst volume. The temperature calculation from the enthalpy

value will require an iteration procedure, in this case lesser complicated than in

equilibrium conditions.

Under equilibrium conditions, subsequent volumes will consider that the molar

fraction of the products of the previous volume will act as the inlet propellant

mixture, so the Minimum Gibbs energy method can be applied to any combustor

volume.

(yk_eq, Teq)i = fminGibbs(Nk,i−1, hi − v2i /2, Pi)

The eUective combustion gas constants (Ri, Cpi, λi, µi) will be derived using

the mixture properties equations as a function of yk_eq,1, see Section 3.1.5. The

pressure is obtained from the perfect gas state equation:

Pi = ρi ·Ri · Ti · η (3.38)

where η is the combustor eXciency. Note that the pressure equation and the

Minimum Gibbs function become an algebraic loop.

Finally, the molar fraction of the products of the last volume will be transmitted

to the outlet port to be used by the Nozzle component or in another possible

chamber.

The term qw appearing in the energy conservation equation permits the ex-

change of convective and radiative heat through a thermal port. The walls (that

can be represented by thermal components or by the Cooling Jacket component)

are not included in this component:

qw = hcAwet (Taw − tp.T ) + σAwet (T 4 − tp.T 4) (3.39)

tp is the name of the thermal port (with n nodes in axial direction) connected to

58


the Combustor. σ is the Stefan-Boltzmann constant = 5.67.10-8 [W∆m−2∆K−4].

The heat exchanged with the Wuid is transmitted through this port: tp.q(i) = qw,i;

tp.T(i) behaves as the internal wall temperature, to be determined in the connected

wall component. Taw is the adiabatic wall temperature deVned as:

Taw = T

(1 + Pr0.33

ref

γ − 1

2M2

); Prref =

(Cpλ

µ

)ref

The reference conditions (ref) are calculated at a temperature halfway between

the wall and the free stream static temperature. It is supposed that the mixture

composition do not change between these two temperatures. The Vlm coeXcient

for each volume is calculated using empirical correlations according to Bartz [9]:

hc = 0.026µ0.2ref

(λref

µref

)0.6

Cp0.4ref (mth)0.8/A0.9

(πDth

4Rcurv

)0.1

(3.40)

where, µref : viscosity of combusted gases at volume no. i and Tref temperature

λref : conductivity of combusted gases at volume i and Tref temperature

Rcurv: Curvature radius of the throat

Dth: Throat diameter

mth: Throat mass Wow

A: Cross section at volume i.

3.4.3. Combustor rate

This component represents non-equilibrium, non adiabatic quasi 1-D combus-

tor component for liquid or gas propellants. The transient chamber conditions

(pressures, temperatures, mass Wows and heat exchanged with the walls) will

be derived from general quasi 1-D transient conservation equations. The non-

equilibrium model is only a Vrst approach; it does not include Vnite rate chemistry

but only time-delay parameters.

The mass, energy and momentum equations include those of the equilibrium

Combustor component plus the ones concerning the vaporization model. Liquid

phase conservation equations are also included.

59


Gas mixture mass conservation equation:

A∂ρ

∂t+∂ρvA

∂x=mvap_fuAfu

Vfu+mvap_oxAox

Vox(3.41)

Gas mixture momentum conservation equations:

A∂ρv

∂t+∂[(ρv2 + P )A]

∂x= −1

2

dξ

dxρ v|v|A+ P

(dA

dx

)(3.42)

Gas mixture energy conservation equation:

A∂ρE

∂t+∂ρvHA

∂x=dqwdx

+qvap_fuAfu

Vfu+qvap_oxAox

Vox(3.43)

where, ρ and E are the gas mixture density and total energy; v is the mean

velocity; mvap_ox andmvap_fu represent the vaporized liquid mass Wow of oxidizer

and of the reducer.

At injector level, the gas mass and enthalpy Wows (mjun,0) will be calculated

by the Injector components taking into account the quality calculated in the

Cavities components. The liquid contributions will be considered in the droplets

conservation equations.

mfu = mfu,inj xfu; mox = mox,inj xox

where mox,inj , mfu,inj are the injectors’ mass Wow (liquid or gas) and xfu and

xox are the vapour mass fractions in the cavities. Then mjun,0 = mfu + mox. An

hyperbolic tangent term is added to produce a relaxation of the injected vapour

mass Wow at very low temperatures.

Since a centred scheme with a staggered mesh is adopted, total enthalpies hjun

are calculated using the upstream cell conditions: hjun,i = hi−1 = (u+ P/ρ)i−1.

Starter terms are in included in the same way as in the combustor_eq compo-

nent. Also, the node velocities v are calculated as in the combustor_eq component.

The burned gases production rate does not contribute explicitly to the gas

mixture equation because the mass is conserved in the chemical reactions (no

condensation).

60


Vaporization Wows, mvap, and enthalpies Wows, qvap (source term for the gas

mixture conservation equations), are calculated by the droplet vaporization models.

Two models are available:

User defined model :

The vapour mass Wows in each volume are calculated assuming a characteristic

vaporization time modulated with user deVned vaporization factors:

mvap_fu, = fvapMliq_fu/τvap; mvap_ox = fvapMliq_ox/τvap

qvap_fu = mvap_fuh(Tliq_fu); qvap_ox = mvap_oxh(Tliq_ox)

where, Mliq is the liquid mass at the volume i; Tliq are the liquid droplets tem-

peratures, τvap is the characteristic vaporization time and fvap represents the

vaporization factor (time dependant input data) at the volume i.

Droplets vaporization model :

Assuming a very thin saturated layer between the droplets and the surrounding

gases, the conservation equations establish that the sum of convective heat plus

enthalpy mass Wow are the same at both sides of the layer.

Then, the following set of equations is applied at each combustor volume i,

allowing the calculation of the mass and energy exchanges through this layer:

mvap_fu = Aliq_fu

(hc(T − Tsat_fu) + hc,liq_fu (Tliq_fu − Tsat_fu)

)(hvap_fu − hliq_fu)

qvap_fu = mvap_fu hvap_fu − Aliq_fu−gas hc (T − Tsat_fu)

hc,liq_fu = 2λliq_fu/Ddroplet_fu

Aliq_fu = fvap 6Mliq_fu/ρliq_fu/Ddroplet_fu

61


mvap_ox = Aliq_ox

(hc(T − Tsat_ox) + hc,liq_ox (Tliq_ox − Tsat_ox)

)(hvap_ox − hliq_ox)

qvap_ox = mvap_ox hvap_ox − Aliq_ox−gas hc (T − Tsat_ox)

hc,liq_ox = 2λliq_ox/Ddroplet_ox

Aliq_ox = fvap 6Mliq_ox/ρliq_ox/Ddroplet_ox

where, T are the gas temperatures, hc is the heat exchange coeXcient at gas side.

Same value as for the wall is used. Tsat is the saturation temperature calculated

at the partial vapour pressure of volume i; hvap and hliq represent the saturation

enthalpies calculated at the partial vapour pressure of volume i;

Ddroplet is the mean droplets diameter and Aliq is the equivalent exchange area

between the droplets and the gas.

We point out that the droplet diameter has been modulated by the vaporization

factor. This factor is an input data depending on the time and on the volume

number.

In theory, assuming a known droplet size at the injection plate, the droplet

diameter evolution could be determined by “simple” equations relating the evapo-

rated mass Wow with the liquid mass conservation equations. Nevertheless, due

to the high penetration and break up of the liquid jets, it seems more realistic to

assume a known droplet size at each chamber volume, the number of droplets

being determined by the current liquid mass.

In both methods (used deVned and droplet model), the vapour mass Wow is also

weighted with a factor to prevent vaporisation at very low temperatures (frozen

liquid):

mvap = mvap tanh

(10T − Tsat

Tsat

)In normal situations, T > Tsat, making the value of the hyperbolic tangent be

equal to one.

Burning Rate :

The burned gases mass Wow (second source term for the vapour conservation

62


equations, see Eq. 3.44a and Eq. 3.44b) is calculated assuming a global characteris-

tic burning time [9]. It is also supposed that any species (vapour or burned gas)

present in gas mixture contributes to the global reaction rate, so the burning rate

will be proportional to the total gas mixture density:

mbu = fbuρ V

τbu

where τbu is the characteristic burning time and fbu is the burning factor at the

volume i.

The burning factors are automatically set to one if the burning conditions are

true: mixture ratio within the allowed limits and ignition Wag activated. Otherwise

the burning factors are set to zero.

vapours / non-condensable / solid propellant gases mass equations :

The vapours and non-condensable mass conservation equations take into account

the burned gases production and the vaporization terms previously calculated:

∂(xfuρ)

∂t+∂(xfuρvA)

∂x=mvap_fuAfu

Vfu− mbu

xfuAfu

Vfu(3.44a)

∂(xoxρ)

∂t+∂(xoxρvA)

∂x=mvap_oxAox

Vox− mbu

xoxAox

Vox(3.44b)

∂(xncρ)

∂t+∂(xncρvA)

∂x= 0 (3.44c)

∂(xpwρ)

∂t+∂(xpwρvA)

∂x= 0 (3.44d)

The mass Wow of the injected vapours (mfu, mox) and injected non-condensable

gases will be added as source terms to the respective conservation equation of

the Vrst chamber volume. In the same way, the mass Wow of the solid propellant

gases, starter_m, will be added as source a term to the solid propellant gases

conservation equation of the Vrst chamber volume.

63


Liquids conservation equations :

The liquid mass and enthalpy Wows are calculated assuming that vgas = vliq and

neglecting Cpliq derivatives:

∂Mliq_fu

∂t+∂(Mliq_fu v)

∂x= −mvap_fu

∂

∂t(M T )liq_fu +

∂

∂x[(M T )liq_fuv] = −qvap_fu/Cpliq_fu

∂Mliq_ox

∂t+∂(Mliq_ox v)

∂x= −mvap_ox

∂

∂t(M T )liq_ox +

∂

∂x[(M T )liq_oxv] = −qvap_ox/Cpliq_ox

Combustion gases properties calculation :

It is supposed that any molar fraction follows a global reaction rate accordingly

with the previously mentioned burning time:

dyk,bu

dt=

(yk,eq − yk,bu)

τbu

where yk,eq is the equilibrium molar fraction of the chemical constituent k at each

volume i. yk_bu is the actual burned molar fraction of the chemical constituent k

at each volume i, and τbu is characteristic burning time.

For each chamber volume i, the combustion gases equilibrium composition

(needed for the calculation of the actual burned gases composition) is calculated

using the Minimum Gibbs energy method as a function of the gas mixture molar

fractions, pressure and the enthalpy.

The gas mixture molar fractions in each volume are calculated as follows:

Reducer vapours contribution:

Nk = xfuyk,fu

MWmix,fu; MWmix,fu =

Nchem∑k=1

yk,fuMWk,fu

64


Oxidiser vapours contribution:

Nk = Nk + xoxyk,ox

MWmix,ox; MWmix,ox =

Nchem∑k=1

yk,oxMWk,ox

Solid propellant gases contribution:

Nk = Nk + xpwyk,pw

MWmix,pw; MWmix,pw =

Nchem∑k=1

yk,pw MWk,pw

Non-condensable gases contribution:

Nnc = Nnc +xnc

MWnc

Burned gases contribution:

Nk = Nk + xbuyk,bu

MWmix,bu; MWmix,bu =

Nchem∑k=1

yk,buMWk,bu

The vapour (reducer and oxidizer mixtures) molar fractions have been calculated

by the cavity components and can include any allowed Wuid mixture using the

fluid_flow_1d library, including that of a previous combustion. The burned

molar fractions, yk_bu, are calculated dynamically.

Once the number of moles of the reducer/oxidizer/burned gases mixture has

been evaluated, and using the “dynamic” enthalpy value obtained from the con-

servation equations, it is possible to call to the Minimum Gibbs energy method to

obtain the equilibrium combustion gases composition:

(yk_eq, Teq)i = fminGibbs(Nk,i, hi − v2i /2, Pi)

Two possibilities are foreseen calling the previous function: Equilibrium and

frozen Wow. In the last case (no ignition) the molar fractions remain constant:

yk_eq,i = (Nk,i).

The eUective combustion gas constants (Ri, Cpi, λi, µi) will be derived using

the mixture properties equations as a function of yk_bu,i see Section 3.1.5.

The pressure is obtained from the perfect gas equation: Pi = ρi ·Ri · Ti. With

65


respect to the Combustor_eq component, the pressure equation and the Minimum

Gibbs function do not become an algebraic loop because the actual molar fractions

(yk_bu,i) of the mixture are dynamic variables.

The molar fraction of the products of the last volume will be transmitted to the

outlet port to be used by the Nozzle component or in another possible chamber.

3.4.4. Nozzle

This component represents 1D supersonic nozzle in quasi-steady conditions. Two

possibilities are foreseen in the main body of this component:

• an “ideal” nozzle using a variable gamma approximation

• a non-adiabatic, non-isentropic nozzle

The mathematical model explained below can be applied either to a complete

nozzle or to a nozzle extension. The important point is to know the inlet total

conditions calculated from the upstream enthalpy/entropy conditions and from

the mass Wow calculated in the complete nozzle component:

- If the nozzle is connected to a chamber (case of a complete nozzle), then the

total conditions will be those of the exit of the chamber transmitted by the

nozzle port.

- If the nozzle is connected to another nozzle (case of a nozzle extension),

then the total conditions will be those of the exit of the upwind nozzle.

Inlet/outlet mass Wows are the same.

The throat calculation (choked mass Wow, see below) will be only implemented

in the complete nozzle.

The choked throat conditions (Pth, Tth and vth) can be calculated with the

following three equations assuming that the total enthalpy and the static entropy

are known (those of the last combustor volume):

hch,tot = h(Nk,th, Tth) + v2th/2

sch = s(Nk,th, Pth, Tth)

vth =√γthRth Tth; mth = Ath vth Pth/(Rth Tth)

66


where, γth is the isentropic coeXcient. This is a function of Pth, Tth, Nk,th; s is the

entropy, a function of Pth, Tth and Nk,th (see section 3.1.1); h is the static enthalpy.

This one is a function of Tth and Nk,th (see section 3.1.1).

Nk,th is the number of moles of the chemical constituent k of the burned gases

at throat. It is calculated from the Minimum Gibbs energy method:

(Nk, T )th = fminGibbs(Nk,ch, sch, Pth)

where sub index “ch” denotes the conditions at the exit of the combustor. Two

possibilities are foreseen calling the previous function: Equilibrium and frozen

Wow. In the last case, molar fractions remain constant. The equations above are

solved iterating in pressure and in temperature.

To cover subsonic conditions, the eUective throat mass Wow is calculated using

a dynamic momentum equation (see Section 3.2.2) where the left pressure cor-

responds to the combustor exit, and the right one to the external pressure. Of

course, under normal steady conditions, the dynamic mass Wow will be limited to

the critical mass Wow, mth, previously calculated.

“Ideal” supersonic nozzle :

The nozzle is divided in sections. Assuming an isentropic frozen expansion

between the node i and i+1 the temperature and pressure for each section can be


θi =Ttot

Ti= 1 +

(γi − 1

2

)M2

i ; δi =Ptot

Pi= θ

γiγi−1 (3.45)

where, γi is the burned gases isentropic coeXcient at section no.i. This is a

function of T

Mi is burned gases Mach number at section no.i.

Ttot is the burned gases total temperature (input).

Ptot is the burned gases total pressure (input).

The closing condition to calculate the Mach number knowing the area ratio is:

MiAi

Ath=

(2θiγi + 1

) γi+1

2(γi−1)

; γi = f(Ti) (3.46)

67


The equations above are solved iterating in Mach and in temperature.

Non-adiabatic, non-isentropic supersonic nozzle :

It is assumed that the non-adiabatic process takes place into two separated steps:

Heat losses :

From section i to i+1, the following losses in enthalpy and entropy will take place:

htot,i+1 = htot,i − qw,i/mth (3.47)

si+1 = si − qw,i/mth/Ti (3.48)

where qw,i is calculated with Equation 3.39 (using Bartz correlations). For the Vrst

nozzle station (i = 1) the total enthalpy and the static entropy are known (those

of the exit of upstream component).

Expansion :

Assuming now that the isentropic relations are valid we have:

htot,i+1 = h(Nk,i+1, Ti+1) + v2i+1/2 (3.49)

si+1 = s(Nk,i+1, Pi+1, Ti+1) (3.50)

mth = ρi+1 vi+1Ai+1 (3.51)

Nk,i is the number of moles of the chemical constituent k at the nozzle sta-

tion i. It is calculated from the Minimum Gibbs energy method: (Nk, T )i+1 =

fminGibbs(Nk,i, si, Pi+1).

The three equations above with three unknowns (pressure, temperature and

speed) are solved iterating in pressure and temperature. Two possibilities are

foreseen calling the previous function: Equilibrium and frozen Wow. In the last

case, molar fractions remain constant.

The following numeric derivatives are done to calculate the isentropic coeXcient

68


at equilibrium conditions:

∂ ln ν

∂ lnT= 1 + lnNtot,dT / ln(1 + ε)

∂ ln ν

∂ lnP= 1 + lnNtot,dP / ln(1 + ε)

where Ntot,dT , Ntot,dP are the total number of moles (increments with respect to

one) due to separated perturbations in P/T (∆P = εP ; ∆T = εT, ε = 1e− 4):

MinGibbsEnergy_PT(mix, Nk, T (1 + ε), P, Nk,dT , Ntot,dT

)MinGibbsEnergy_PT

(mix, Nk, T, P (1 + ε), Nk,dP , Ntot,dP

)Nk is the number of moles of the chemical constituent k at equilibrium con-

ditions. Nk,dT , Nk,dP are the number of moles after perturbation in T, P. The

calculation of the speciVc heat at constant pressure, Cp, has two terms: Cpfr and

Cpre. The frozen one is calculated as in Section 3.1.1 and the reactive term is


Cp = Cpfr + Cpre

Cpre =∑

k

(∂ lnNk

∂ lnT

)·Nh

hk

T MWmix

where, (∂ lnNk

∂ lnT

)=

(ln(Nk,dT /MWmix,dT )− ln(Nk/MWmix)

)ln(1 + ε)

Then calculation of the speciVc heat at constant volume and the isentropic

coeXcient is:

Cv = Cp+R

MWmix· (∂ ln ν/∂ lnT )2

∂ ln ν/∂ lnP

γ =Cp

Cv (∂ ln ν/∂ lnP )

Under frozen conditions, (∂ ln ν/∂ lnP ) is equal to one.

Regarding the heat exchanged with the walls, it is used the same formulation

present in the Combustor model (see Equation 3.39). In the case of an ideal nozzle

it is supposed that the heat exchanged with the walls is a small quantity with

69


respect to the enthalpy Wow, so the isentropic equations are supposed still valid.

In order to calculate the thrust of the engine, here below the expressions needed

to calculate the thrust F and the Isp at section i:

Fi = mthMi

√γiRgas Ti + (Pi − Pout)Ai

ISPi = Fi/mth

where Pout is the external boundary condition in pressure; Rgas represents the

burned gases constant at nozzle exit m is the mass Wow at throat.

3.4.5. Cooling Jacket components

These components represent a Regenerative Circuit of a Chamber. Two models

are here described: the CoolingJacket component and the CoolingJacket_simple

component. For the Vrst one a 3-D geometry (built by means of several 3-D walls

around the channels) is taken into account. For the second one, a simpliVed wall

geometry is considered.

Cooling Jacket component

It is constructed by aggregation of one Tube (fluid_flow_1d library) representing

the channels and Vve 3-D walls around them. The cooling jacket is divided into a

variable number of sections in axial direction. Every section is made of:

• one Wuid node of the Tube component (fluid_flow_1d library), which is

simulating the cooling channels

• Vve slides of the wall_3D components, which are simulating the metallic

walls. They are arranged according to Figure 3.5

Channels: they are simulated by only one Tube component making its “num”

data equal to the number of channels. The mesh size of each node is function of

the axial position through the non-dimensional geometry tables.

70


Figure 3.5.: Cooling jacket wall mesh [43]

The rectangular channel geometry (widths, heights, wet areas) is similarly

calculated but using the interpolated widths and heights values as follows:Awet,i = 2 · (ai + bi) · lch,i

ai = wch · interp(xi/L, wc_vs_L)

bi = tch · interp(xi/L, tc_vs_L)

Heat conduction: the wall_3D components used to simulate the walls will

calculate the heat conduction in every direction including the axial direction.

This thermal component features thermal ports in radial and in azimuth di-

rections allowing an exact calculation of heat conduction through the channel

corners. The walls are divided in 5 diUerent 3-D components as shown in Fig-

ure 3.5. Each component has a 3-dimensional discretisation in tangential, radial

and longitudinal direction (dx, dy, dz), respectively.

The formulation for this component is the typical one for 3-dimensional con-

duction elements; the thermal capacitance for each volume is deVned as:

Ci,j,k = ρCp(i,j,k) dx dy dz (3.52)

71


the internal heat Wows are evaluated by:qx(i,j,k) = ki,j,k dy dz (Ti−1,j,k − Ti,j,k)/dx

qy(i,j,k) = ki,j,k dx dz (Ti,j−1,k − Ti,j,k)/dy

qz(i,j,k) = ki,j,k dx dy (Ti,j,k−1 − Ti,j,k)/dz

(3.53)

while the energy equation is:

Ci,j,kdTi,j,k

dt= qx(i,j,k)−qx(i+1,j,k)+qy(i,j,k)−qy(i,j+1,k)+qz(i,j,k)−qz(i,j,k+1) (3.54)

As shown in Figure 3.5 only half channel has been considered because of

symmetry reasons, with left and right sides adiabatic:qout,right_r = 0

qout,int_l = 0 qout,int_right_r = 0

qout,ext_l = 0 qout,ext_right_r = 0

The heat Wux to the external side is calculated as the sum of all wall nodes

connected to the ambient:

qamb(i) = 2 · nch

N∑j=1

qout,ext(j) +N∑

j=1

qout,ext_right(j)

The temperatures of the external walls (for each section i) in contact with the

exterior are supposed to be the same:Tw,amb(i) = Tout,ext(j)

Tw,amb(i) = Tout,ext_right(j)

The same procedure is applied for the heat Wuxes related to the internal side,

the combustion chamber:qin(i) = 2 · nch

N∑j=1

qin,int(j) +N∑

j=1

qin,int_right(j)

Tw,in(i) = Tin,int(j)

Tw,in(i) = Tin,in_right(j)

72


The wet surfaces of the walls (wall_int, wall_right and wall_ext) in contact

with the channel coolant, are supposed to be at three diUerent temperatures (one

for each side): Tch,in = Twall,int

Tch,lat = Twall,right

Tch,out = Twall,ext

The corresponding heat Wux (for each section i) between the channel and the walls

around it are calculated as follows:

qch,in = 2N∑

j=1

qwall,int(j)

qch,lat = 2N∑

j=1

qwall,right(j)

qch,out = 2N∑

j=1

qwall,ext(j)

The channel heat Wuxes are also calculated by the Tube component using as

input the wall/Wuid temperatures and two phase hydraulic correlations for the

Vlm coeXcient evaluation. Then, a set of implicit non linear equations is formed

which is solved by EcosimPro. This also applies for the combustion chamber side,

where the heat Wuxes are calculated by the Combustor component connected to

the CoolingJacket component.

Concerning the external side (ambient), the heat Wow shall be deVned according

to the thermal component connected to this port.

Cooling Jacket simple component

This component is constructed by aggregation of one Tube (fluid_flow_1d

library) representing the channels and three 1D bars around them.

The philosophy is the same as in the CoolingJacket component with the dif-

ference that here the Vve 3D-walls are replaced by three 1D-bars with only one

thermal node per section. So, the cooling jacket is divided into a variable number

of sections in axial direction. Every section is made of:

• one Wuid node of the Tube component (fluid_flow_1d library), which is

73


Figure 3.6.: SimpliVed Cooling Jacket wall disposition [43]

simulating the cooling channels

• three slides of the bar_1D components, which are simulating the metallic

walls. They are arranged according to Figure 3.6

Figure 3.6 shows the diUerent temperatures and heat Wuxes within the cooling

channel. There Tc is the temperature of the hot combustion gas, TW is the temper-

ature of either the combustion chamber - or nozzle wall, TR is the temperature of

the cooling rib, TCH is the temperature of the cooling Wuid, TS is the temperature

of the surface and Tamb is the ambient temperature. For the model it is assumed

that the temperatures are constant over the separate volumes.

Heat always Wows from regions with a higher potential to regions with a lower

potential. Starting with the heat Wux QCC,W from the hot combustion gas to the

chamber - or nozzle wall, the heat Wux then splits up in a smaller part which goes

into the channel ribs QW,R and the main part which goes into the cooling Wuid

QW,CH . From the ribs the main part Wows into the cooling channel QR,CH , but

also a small part Wows to the surface QR,S .

The heat from the channel is carried downstream with the Wuid, but a small

fraction might Wow to the surface QCH,S . The surface itself transfers heat to

74


Figure 3.7.: Channel with relevant areas and surfaces for heat Wux calculation

the surrounding Qrad. There QCC,W and Qrad transfer heat by radiation and

convection. For the calculation of QW,CH , QR,CH and QCH,S one solely has to

consider the convective heat transfer. The heat Wuxes QW,R and QR,S within the

solid material are based on the conductive heat transfer.

Prior to the listing of the heat Wuxes the geometry of the cooling channels are

explained. Hence, the cross-sectional area of the cooling channel is given once

more in Figure 3.7. Additionally, the relevant areas for the heat Wux calculation

are entered.

Aint, Arib, Aext andAch describe the cross-sectional areas of the wall, the cooling

rib, the surface and the cooling channel. They can be calculated as follows:

Aint =

(πD

n+ wch

)ti4

Arib = wr2tc + ti + te

2

Aext =

(πD

n+ wch

)te4

Ach = wc · tc

75


Sw, Sw,r, R, Sch,int, Sch,rib, Sch,ext, Sr,s, and Sext describe the intermediate

surfaces between the internal space of the combustion chamber or nozzle, the

wall, the cooling rib, the cooling Wuid, the surface and the surrounding. They can

be calculated as follows:

Sw = Sext = l · (wch + 2wr)

Sw,r = l ·√w2

r + t2i

Sr,s = l ·√w2

r + t2e

Sch,int = Sch,ext = l · wch

Sch,rib = l · tch

By means of these values, the equations for the heat Wuxes and the energy

conservation can be set up. In the following the heat Wuxes and subsequent the

equations for the energy conservation are listed for a segment i:

qwall = hcAwet (Taw − tp.T ) + σAwet (T 4c − tp.T 4)

qcha =lπD

2n

λint

ti/2(tp.T − Tw,int)

qcoo,int =lwc

2

λint

ti/2(Channel.tpin.T − Tw,int)

qcoo,ext =lwc

2

λext

te/2(Channel.tpout.T − Tw,ext)

qcoo,rib = lwcλint

wr/2

(Channel.tplat.T − Tw,rib

)qrib,int = Sw,r

λint

(ti + tc + wc)/2

(Tw,rib − Tw,int

)qrib,ext = Sr,s

λext

(te + tc + wc)/2

(Tw,rib − Tw,ext

)

(3.55)

where λint and λext are the heat conductivities of the internal walls/ribs and

external walls, respectively. Please note that the heat Wux qwall equals the heat

Wux from the combustion chamber (Eq. 3.39).

Additionally to the radial heat Wuxes (Equations 3.55), the longitudinal heat

Wuxes need to be regarded. Since the cooling channel has four solid cross-sectional

areas (AS, 2 x AR, AW), four longitudinal heat Wuxes are calculated at every

junction. Figure 3.8 illustrates these heat Wuxes for a segment i.

76


Figure 3.8.: Longitudinal heat Wuxes for a segment i

The longitudinal heat Wuxes are based solely on the heat transfer mechanism

conduction. In the following the associated equations are given:qz,int(i) = Aint

2λint

li−1 + li(Tw,int(i− 1)− Tw,int(i))

qz,ext(i) = Aext2λext

li−1 + li(Tw,ext(i− 1)− Tw,ext(i))

qz,rib(i) = Arib2λint

li−1 + li

(Tw,rib(i− 1)− Tw,rib(i)

) (3.56)

In order to calculate the energy conservation of the cooling jacket, one has to

regard the diUerent volumes separately. Subsequent the equations are given for

the wall, the cooling rib and the surface:

dTw,int(i)

dt(ρint · Vint)iCp(i) = qcha(i)− qcoo,int(i)− qrib,int(i) + qz,int(i)− qz,int(i+ 1)

dTw,ext(i)

dt(ρext · Vext)iCp(i) = −qext(i)− qcoo,ext(i)− qrib,ext(i) + qz,ext(i)− qz,ext(i+ 1)

dTw,rib(i)

dt(ρint · Vrib)iCp(i) = qrib,int(i) + qrib,ext(i)− qcoo,rib(i) + qz,rib(i)− qz,rib(i+ 1)

(3.57)

77

4. Steady State Library

The already available propulsion library ESPSS can be used to study both station-

ary states and transients of a propulsion system. Unfortunately its use for steady

state applications is not trivial because of the complexity of the transient models

there implemented. Therefore, simpliVed pre-design and parametric studies are

diXcult and time-consuming. The library shown hereafter is designed speciVcally

for steady state purposes, providing a helpful and fast tool for the pre-design phase

(feasibility analysis) allowing for parametric studies.

To this aim, the available Wuid properties and combustion modelling functions

of ESPSS have been implemented in an adequate form into new libraries. Addi-

tionally, Wuid dynamic, combustion and heat transfer models have been developed

to simulate the physical steady state behaviour of the main components of a

propulsion system, as pipes, valves, turbines, pumps, oriVces, combustion cham-

ber and nozzle. These components are suited for both launcher and spacecraft

applications.

4.1. Components Overview

The set of models developed for the State state library are able to represent most of

the components of a liquid propulsion system for spacecraft or rockets. Figure 4.1

shows an overview of the components model developed for the steady state

library.

4.2. Ports

Ports are used to connect components to each other, in order to guarantee the

propagation of the variables. Two new ports have been created: the Steady

State Wuid port [34], that represents the basis and the rationale on which the

78


Figure 4.1.: Components in the Steady State library

79


whole library is coded, and the nozzle port, to better manage nozzle connections,

similarly to the transient nozzle port.

Each port can connect two or more components at once. SUM variables, such

as mass Wow rate, will be summed at the ports to ensure mass Wow conservation;

EQUAL variables, such as stagnation pressure, will be propagated to all compo-

nents connected to the same port. A standard Wow component should have two

ports, one IN and one OUT, thus deVning the mass Wow direction. IN ports ensure

calculation of the enthalpy Wow mh, while enthalpy h is computed in the OUT

ports.

Similarly to the original ESPSS Wuid port, the Steady State Wuid port can

propagate the molar fraction of chemical species to allow the correct evaluation of

Wuid Wow of combustion products in systems where this is required (e.g. staged

combustion cycles).

4.3. The “type” switch

Most components have a “type” switch, that enable switching the model between

Design and OU-Design mode.

• Design mode. Geometrical construction data for junctions and valves are

an output of the components. They are calculated from a given ∆P .

The combustion chamber requests chamber pressure, mixture ratio MR and

throat diameter as main design inputs. The propellant mass Wows and heat

Wuxes are its main outputs.

Turbomachinery components in design mode evaluate performance parame-

ters as torque, power and rotational speed using the mass Wow and pressures

coming from the ports.

• OU-Design mode. This mode can be used for the analysis of a given cycle

with Vxed geometry and main characteristics. Here, junctions and valves

have a given geometry. Mass Wow or ∆P are calculated, depending on the

relative placement of the components.

Combustion chambers have a given nozzle throat diameter (as in design

80


mode), but mass Wows are given from the inlet ports. Chamber pressure and

MR are calculated accordingly.

From this general description some additional options are given in Design

mode, in order to Vne-tune the models depending on the cycle studied. Therefore,

mass Wow ratios through splits can be user-Vxed (inputs), or calculated as outputs.

Likewise, turbine pressure ratios can be Vxed or calculated from the given cycle.

4.4. 1-D pipes

Tubes components are able to evaluate a one-dimensional Wow in steady state

conditions. The tube takes into account the enthalpy variation due to external

heat Wuxes and the pressure drop due to the friction along the pipe. As in the

transient version of the component, the steady state tube is divided in volumes

and junctions. Pressure drops and enthalpy variations are calculated at the end of

each volume, on the junction.

The governing equations based on the one-dimensional steady state model are

the following:

Mass conservationd

dx(ρvA) = 0 (4.1)

Momentum conservation

dP

dx+ ρv

dv

dx+

1

2ρv2 fr

Di= 0 (4.2)

where the friction factor fr is a function of Re and relative roughness, as deVned

in Ref. [43].

Energy conservation

ρvAdh0

dx= hc (Tw − T )Pw (4.3)

81


Variable Description UnitInputsm Mass Wow rate [kg/s]h1 Inlet enthalpy [kJ/kg]P2 Outlet Pressure [Pa]ParametersPo Initial pressure [Pa]To Initial temperature [K]rhoo Initial density [kg/m3]mo Initial mass Wow [kg/s]rug Roughness [m]kf Multiplier of the friction factor [-]alphabend Bend angle [deg]Rbend Ratio of curvature bend [m]Wdadd Additional losses in fL/D [-]ht_option Heat transfer option [-]L Tube length [m]D Nominal tube inner diameter [m]

Table 4.1.: 1-D pipe element

The heat Wux is evaluated using the thermal port and connecting the com-

ponent with all components inside the EcosimPro thermal library. It has been

decided to use the original transient EcosimPro thermal library, but still allowing

to be interfaced with the steady state library. This choice enables a relaxation

in the overall steady state model stiUness thanks to the Vrst order diUerential

equations present in the thermal components. Of course, due to the speciVcations

of the steady state library, the time variable will have no physical meaning

anymore, and it must rather be regarded as an integration constant.

The pipe component is inherited from the tube. Additionally it features a

1-D wall model for the evaluation of the heat Wuxes and heat capacities. As

in the corresponding transient component, it includes a material pipe thermally

connected to the tube, and permits simple convection with the ambient by using a

constant convective coeXcient hc.

82


4.5. 0-D components: junctions & valves

0-D components represent concentrated pressure loss in a propulsion system, such

as oriVces and valves. These components are based on a similar model; the valve

component diUers from the oriVce only because it enables the variation of the

throat area while the oriVce presents a constant one. Both the components feature

a Design and an OU-Design mode.

Variable Description UnitInputsm Mass Wow rate [kg/s]h1 Inlet enthalpy [kJ/kg]P2 Outlet Pressure [Pa]Outputsh2 Outlet enthalpy [kJ/kg]P1 Inlet Pressure [Pa]A Valve Area [m2]Parameters∆P Pressure loss (Design mode) [Pa]ζ Pressure drop coeXcient [-]mo Initial mass Wow [kg/s]

Table 4.2.: 0-D Junction element

In the Design mode the pressure drop is not related to the mass Wow but is an

input coming from the ports or from the user and the geometric parameters of the

junction/valve are calculated:

Kjun =2∆Pρ

m2Kvalve =

2∆Pρ

m2 pos2

A =

√ζ

K

In the second case, the OU-Design mode, the model calculates the concentrated

pressure drop as:

∆P = ζm2

2ρA2(4.4)

83


and the mass Wow is calculated implicitly from Equation 4.4.

4.6. Combustion Chambers

The thrust chamber is composed by two diUerent components following the same

idea developed in the transient library. A combustor component and a nozzle

component are linked together to create the thrust chamber.

Combustor

This component represents a non adiabatic 1-D combustion process inside a

convergent chamber (up to the throat section). It is a steady state, one dimensional,

isoenthalpic combustion chamber. The equilibrium combustion products are

calculated using the minimum Gibbs energy method [55] as a function of the

propellant’s mixture molar fractions and enthalpies and of the chamber pressure.

The chamber geometry allows for precise chamber contour deVnitions and non

homogeneous node discretisation.

Thermodynamic properties along the chamber sections are evaluated using

isoenthalpic correlations in frozen conditions. Heat Wuxes are calculated with

the Bartz correlation in closed form [9]. The chamber works only in “ignited”

mode, as it is not required for a steady state model to show transitions between

non burning and burning state. The compositions of the combustion products

are evaluated from the injected Wuids. It is possible to use either pure Wuids or

combustion products from a previous combustor (preburner). Along with the

Design/OU-Design type switch, another switch is responsible for choosing the

combustor type, which can be either aMain Combustion Chamber (MCC) or a

Preburner (PB_GG). The main diUerence between the two combustor types is the

following:

- MCC. For a Main Combustion Chamber in Design mode, chamber pressure

is user given (Pc = Pdesign)

- PB_GG. For a Preburner in Design mode, chamber pressure is taken from

the outlet port (Pc = f_out.P )

84


Variable Description UnitInputshox Oxidiser Inlet enthalpy [kJ/kg]hfu Fuel Inlet enthalpy [kJ/kg]Outputsm Mass Wow rate [kg/s]mox Oxidiser mass Wow rate [kg/s]mfu Fuel mass Wow rate [kg/s]Pc Chamber pressure [Pa]Tc Chamber temperature [K]Tw Chamber wall temperature [K]ParametersPco Chamber pressure [initial if OU-D; assigned if D] [Pa]Tco Initial combustion temperature [K]Tcox Initial Oxidiser combustion temperature [K]Tcfu Initial Fuel combustion temperature [K]MRo Mixture Ratio [initial if OU-D; assigned if D] [-]ηc Combustion eXciency [kg/s]Lc Chamber length of subsonic part [m]Dt Nozzle throat diameter [m]

Table 4.3.: Combustor element

85


In Design mode, the combustor component calculates inlet mass Wows from

given chamber pressure, MR and nozzle throat diameter. The mass Wow conserva-

tion is written as:

m =ρthvthAth

ηc(4.5)

where the subscript th refers to throat conditions, and ηc is the combustion

eXciency.

In OU-Design mode, the component evaluates the equilibrium composition in

the Vrst section to obtain thermodynamic and transport properties, and in the

throat to evaluate the chamber pressure and the mass Wow rate by an iterative

loop. This equation is actually used in the overall loop to determine the chamber

pressure Pc implicitly. The ideal gas equation is written twice, for stagnation

chamber and for throat conditions. Isentropic throat conditions are calculated

iteratively assuming shifting equilibrium and variable isentropic coeXcient γ.

For each node i, the relevant characteristics (Mach number Mi, Pi, Ti, ρi,

sound speed vsound,i, vi) are calculated assuming isentropic Wow conditions. This

simpliVcation is acceptable since these variables are only needed for assessing the

heat transfer coeXcient within the Bartz equation.

Nozzle

The component represents a 1-D supersonic nozzle in steady state conditions.

The choked throat conditions (Pth, Tth and vth) are evaluated in the combustor

component and communicated through the nozzle port. Stagnation conditions

are calculated from the throat conditions. Static conditions are evaluated in each

section using isentropic correlations and assuming frozen chemistry. The heat

Wux in each section is evaluated using the semi-empirical correlation of Bartz in a

closed form.

86


Variable Description UnitInputsh Enthalpy at throat [J/kg]s Entropy at throat [J/kg K]P Pressure at throat [Pa]T Temperature at throat [K]ρ Density at throat [kg/m3]Cp SpeciVc heat at throat [J/ kg k]m Mass Wow rate [kg/s]OutputsPi Nozzle pressure proVle [Pa]Ti Nozzle temperature proVle [K]Tw Nozzle wall temperature [K]Isp SpeciVc impulse [s]Thrust Thrust [N]ParametersPco Initial nozzle pressure [Pa]Tco Initial nozzle temperature [K]ηCf Nozzle eXciency [-]Pext External pressure [Pa]Ld Nozzle length of supersonic part [m]Dt Nozzle throat diameter [m]

Table 4.4.: Nozzle element

87


4.7. Cooling Channels

Cooling channels component features two diUerent models depending on the

complexity of the propulsion system:

• cooling jacket

• regenerative circuit

The cooling jacket component is inherited from the tube component. It permits

the modelling of a combustion chamber cooling jacket. Mass Wow, pressure drop

and temperature rise of the coolant are evaluated using the same governing

equations as for the pipes and the tubes (see Equations 4.1, 4.2, 4.3). Since the

direction of the Wow in the Steady State library must be given at the schematic

design stage, two components are foreseen: a co-Wow and a counterWow cooling

jacket. They are constructed by aggregation of one tube representing the channels

and a simpliVed 3D geometry (built by means of several bars around the channels)

around them (see Figure 4.2). The regenerative circuit component features a

pre-deVned pressure drop and a pre-assigned hot gas side wall temperature Tw,hot

proVle along the combustion chamber. Mass Wow value is coming from the ports;

the component sends the Tw,hot variable to the combustion chamber through the

thermal port. In this way it is possible to evaluate the chamber heat Wux qw (see

Eq. 4.6). Using this variable, the enthalpy rise along the channel is evaluated and

so the outlet temperature. Choosing a material for the chamber wall, the chamber

wall thickness is an output of the design (see Eq. 4.7). The channel height is a user

given input for the model, while the channel width is evaluated from the Vxed

number of channels.

qw = hcAwet (Taw − Tw,hg) (4.6)

qw =λ

tAwet (Tw,hg − Tw,cf ) (4.7)

88


Variable Description UnitInputsm Mass Wow rate [kg/s]h1 Inlet enthalpy [kJ/kg]P2 Outlet pressure [Pa]OutputsP1 Inlet pressure [Pa]h2 Outlet enthalpy [kJ/kg]T2 Outlet temperature [K]Tcool Coolant temperature [K]Tw Channel wall temperature [K]qw Chamber heat Wux [W]Parametersnch Number of channels [-]Po Initial pressure [Pa]To Initial temperature [K]rhoo Initial density [kg/m3]mo Initial mass Wow [kg/s]rug Roughness [m]kf Multiplier of the friction factor [-]alphabend Bend angle [deg]Rbend Ratio of curvature bend [m]Wdadd Additional losses in fL/D [-]ht_option Heat transfer option [-]wch Channel widths [m]tch Channel heights [m]thi Jacket inner wall thickness [m]the Jacket outer wall thickness [m]mati Jacket internal material [-]mate Jacket external material [-]L Channel length [m]D Nominal channel inner diameter [m]

Table 4.5.: Cooling jacket element

89


Variable Description UnitInputsm Mass Wow rate [kg/s]h1 Inlet enthalpy [kJ/kg]P2 Outlet pressure [Pa]OutputsP1 Inlet pressure [Pa]h2 Outlet enthalpy [kJ/kg]T2 Outlet temperature [K]Tcool Coolant temperature [K]qw Chamber heat Wux [W]tw Internal wall thickness [m]Parametersnch Number of channels [-]Po Initial pressure [Pa]To Initial temperature [K]rhoo Initial density [kg/m3]mo Initial mass Wow [kg/s]dPdesign Design pressure drop [Pa]Tw Channel wall temperature [K]wch Channel widths [m]tch Channel heights [m]the External wall thickness [m]mati Jacket internal material [-]mate Jacket external material [-]L Channel length [m]D Nominal channel inner diameter [m]

Table 4.6.: Regenerative circuit element

90


Figure 4.2.: Cooling jacket channels wall mesh [43]

4.8. Turbomachinery

4.8.1. Pump

The pump component features a simple model using isentropic relations and

constant, user-given eXciency ηp to calculate pump conditions. The isentropic

enthalpy rise is calculated assuming an isentropic transformation between inlet

and outlet pressure.

The Design type parameter decides whether the pump pressure rise is Vxed

from the ports (in Design mode) or calculated (in OU-Design mode). In both

modes, given a assigned speciVc speed Ns, the shaft rotation speed ω is calculated:

Ns =ω√Q/ns

(TDH/nst)0.75 (4.8)

where ns is the number of suctions, nst is the number of stages of the pump, Q

is the volumetric Wow and TDH represents the actual total dynamic head of the

pump. In Design mode, the inlet entropy s is calculated from inlet pressure Pin

and enthalpy hin. Then, the isentropic enthalpy rise dhis is calculated with an

isentropic transformation, with known inlet and outlet pressures Pin and Pout,

91


Variable Description UnitInputsh1 Inlet enthalpy [J/kg]P1 Inlet pressure [Pa]T1 Inlet temperature [K]m Mass Wow rate [kg/s]OutputsP2 Outlet pressure [Pa]ω Rotational speed [rad/s]τ Torque [N m]W Power [W]ParametersPo Initial pressure [Pa]To Initial temperature [K]mo Initial mass Wow [kg/s]rpmo Initial rotational speed [rad/s]ηp Pump eXciency [-]

Table 4.7.: Pump element

and inlet entropy sin. Finally, the real enthalpy rise dh is calculated as:

dh =dhis

ηp(4.9)

and subsequently, the shaft rotational speed ω and torque τ are then linked with

the actual enthalpy rise by the power balance equation:

ω τ = m∆h (4.10)

In OU-Design mode, the inlet entropy sin is calculated (as for the Design mode)

from inlet pressure Pin and enthalpy hin. The real enthalpy rise dh is given by the

power balance equation, and the isentropic enthalpy rise dhis is given by:

dhis = η · dh (4.11)

Therefore, the outlet pressure Pout is calculated with an isentropic transformation,

with known isentropic enthalpy rise dhis and entropy s. As for the Design case,

92


it is possible to compute torque and power from it (thank to an assigned speciVc

speed Ns).

In the near future the oU-design mode (with assigned speciVc speed) will

be upgraded with the capability of using performance maps for eXciency and

pressure head.

4.8.2. Turbine

The turbine component is a model using isentropic relations and constant, user-

given eXciency ηt to calculate turbine conditions. The isentropic enthalpy fall is

calculated assuming an isentropic transformation between inlet and outlet pressure.

The construction parameter turbine_type decides, in Design mode, whether the

mass Wow m is assigned from the ports (known_mWow) thus calculating the

upstream pressure, or the pressures are assigned from the ports (known_pressures,

known_PI_tt), thus calculating the mass Wows. This switch must be carefully set

depending on the cycle studied:

• turbine_type = known_mWow. For closed cycles, in most cases, the mass

Wow is determined by the preburner (staged combustion) or by bypass valves

(expander), therefore it is given from the ports, and the pressure ratio should

be calculated in the turbine component.

• turbine_type = known_pressures. For open cycles (gas generator) or for

one of the turbines in closed cycles with parallel turbines, the pressures

should be Vxed (known_pressures), and the model will Vnd the mass Wow

that equilibrates the pump power.

• turbine_type = known_PI_tt. For open cycles, especially in the pre-design

phase to have a Vrst attempt result and to facilitate parametric studies

changing the pressure ratio Πtt; the model will Vnd the mass Wow that

equilibrates the pump power.

In all working modes (Design or OU-Design, known_mWow or known_pressures,

known_Pi_tt), torque, shaft rotation speed ω and power are given from the

mechanical port.

93


Variable Description UnitInputsh1 Inlet enthalpy [J/kg]P2 Outlet pressure [Pa]T1 Inlet temperature [K]W Power [W]m Mass Wow rate (if know_mWow) [kg/s]P1 Inlet pressure (if known_pressures) [Pa]PItt Pressure ratio (if known_PI_tt) [-]Outputsm Mass Wow rate (if know_pressures or PI_tt) [kg/s]P1 Inlet pressure (if know_mWow) [Pa]ω Rotational speed [rad/s]τ Torque [N m]ParametersPo Initial pressure [Pa]To Initial temperature [K]mo Initial mass Wow [kg/s]PI_tto Initial pressure ratio [-]rpmo Initial rotational speed [rad/s]ηt Turbine eXciency [-]

Table 4.8.: Turbine element

94


• Design mode

- known_mWow. The mass Wow is used to calculate the real enthalpy

fall (Eq. 4.12).

dh = −Power/m (4.12)

From this value, the isentropic enthalpy fall is calculated with the Vxed

eXciency η (Eq. 4.13).

dhis = dh/η (4.13)

The inlet entropy sin is calculated from the outlet pressure Pout and

the ideal outlet enthalpy hout = hin − dhis. Then, the inlet pressure is

calculated from entropy sin and inlet enthalpy hin.

- known_pressures. The inlet entropy sin is calculated from the inlet

pressure Pin and the inlet enthalpy hin. Then, the isentropic enthalpy

fall dhis is calculated knowing the inlet enthalpy hin, the outlet pressure

Pout and the inlet entropy sin (Eq. 4.14).

dhis = hin − f(sin, Pout) (4.14)

Thereafter, the real enthalpy fall is given by using the turbine eXciency

η (Eq. 4.15).

dh = η · dhis (4.15)

The mass Wow is Vnally estimated from Eq. 4.12.

- known_PI_tt. When this turbine type is chosen, the calculation proce-

dure diUers from the one used in known_pressures only by the inlet

pressure Pin given as an output from Pout times Πtt.

• OU-Design mode. The procedure is roughly the same as the Design mode

case with turbine_type = known_pressures.

4.9. Validation

Several test cases have been performed in order to evaluate the reliability of the

Steady State library. Following a step by step approach, Vrst each component

95


singularly and then more complex systems were validated.

4.9.1. Component validations

The schematics shown hereafter are only the graphical interfaces of mathematical

models where all variables of each component are considered. The tool is able of

calculating the steady state of the mathematical model by solving the non-linear

algebraic equation system that results from the built schematic by means of the

“Newton-Raphson” or the “Minpack” method [40].

In order to have a Wexible and robust tool, each component model is carefully

coded to provide the most suitable variables that can be used to break the non-

linear algebraic loops. The choice of the correct variables that enable the equations

system to converge represents one of the most important achievements of this

work.

Pipeline test case

The purpose of this test case is to validate the Steady State pipe component

and demonstrate its proper function compared to a transient component. The

schematic shown in Figure 4.3 has been also built to check the correct behaviour

of Steady State components in long pipelines. A long pipeline is modelled twice,

with standard ESPSS transient components and with Steady State components.

Pressure drop distribution along the pipeline and mass Wow rates are compared

between the two models.

Please note the absence of the volume between two junctions. Purely capacitive

components are not needed in the Steady State library, and it is possible to chain

multiple resistive components in series.

The schematic describes a series of pipes linked together by junctions. A

pressure diUerence has been imposed between inlet and outlet. The input data

shown in Table 4.9 represent the inputs implemented in each component (steady

or transient). The initial conditions have been taken equal for each pipe, with

atmospheric pressure and low initial mass Wow. These conditions are quite distant

from the solution, and the convergence of the steady state code in this case is an

indicator of its robustness.

96


Figure 4.3.: Schematic of the Pipeline test case. Purple: steady state components.Cyan: transient components

Name Description Value UnitsPin Total Pressure at inlet 50 [bar]Tin Total Temperature at inlet 300 [K]Pout Total Pressure at outlet 30 [bar]Po Initial Total pressure in the pipe 1 [bar]To Initial Total temperature in the pipe 300 [K]mo Initial mass Wow in the pipe (guess value) 0.2 [kg/s]rug Roughness 5e-05 [m]L Pipe length 1 [m]D Pipe internal diameter 0.01 [m]

nodes Pipe nodes discretisation 5 [-]Ao Junction area 7e-05 [m2]ζ Loss coeXcient 1 [-]

Wuid Working Wuid Real H2O [-]

Table 4.9.: Pipeline input data

97


The simulation results are summarized in Table 4.10, showing a very accurate

mass Wow calculation and pressure drop distribution.

Name Value Transient Value Steady State Errorm [kg/s] 1.109 1.109 0.006%

∆P1 [bar] 3.110 3.110 0.002%∆P2 [bar] 3.111 3.111 0.002%∆P3 [bar] 3.112 3.112 0.002%∆P4 [bar] 3.112 3.112 0.002%

Table 4.10.: Pipeline output data

Combustion Chamber

The purpose of this test case is to calculate the main characteristics of combustion

chamber and nozzle components. Its schematic is shown in Figure 4.4 for both

steady state and transient models. The only diUerence between the two models

(besides the diUerent modelling approach) is the absence of the injector capacity

inside the Steady State combustion chamber injection plate.

Relevant input data are listed in Table 4.11. The initial conditions are the same

for the steady state and the transient components.

The test compares the propellant mass Wows and the chamber pressure and

temperatures between the two models. Other important characteristics as heat

Wuxes, wall temperatures and adiabatic wall temperatures have been evaluated as

well, but are not reported here for simplicity.

The output data from the two models are compared in Table 4.12. It is evident

that the steady state results are very similar to the respective transient results.

98


Figure 4.4.: Schematic of Combustion Chamber test case. Purple: steady statecomponents. Cyan: transient components

Name Description Value UnitsPin,ox Ox. Total Pressure at inlet 70.8 [bar]Tin,ox Ox. Total Temperature at inlet 94.7 [K]Pin,fu Fu. Total Pressure at inlet 71 [bar]Tin,fu Fu. Total Temperature at inlet 208.9 [K]Nsub Number of subsonic nodes 5 [-]Nsup Number of supersonic nodes 5 [-]Lcc Chamber length of subsonic part 0.5 [m]Dth Nozzle throat diameter 0.10 [m]Pcc Initial Chamber pressure 1 [bar]Tcc Initial Chamber temperature 300 [K]

Table 4.11.: CC input data

99


Name Transient Value Steady State Value Errormox [kg/s] 18.72 18.53 1.0 %mfu [kg/s] 3.14 3.11 1.0 %mtot [kg/s] 21.86 21.64 1.0 %MR [-] 5.959 5.956 0.04 %Pcc [bar] 64.97 64.27 1.1 %Tcc [K] 3518 3514 0.11 %Mach [-] 2.887 2.762 4.5 %

Table 4.12.: CC output data

4.9.2. Subsystem validations

HM7B Turbopump subsystem

This test case was used during the ESPSS Industrial Evaluation from Astrium

Bremen to validate the ESPSS library for liquid rocket engine cycles [41].

The schematic shown in Figure 4.5 represents the turbomachinery power pack

of the upper stage engine of the Ariane 5 launcher, the HM7B engine, including

the gas generator and both turbopumps.

Figure 4.6 shows the equivalent schematic implemented with Steady State

components. They are very similar to each other. Only volume components and

non-condensable Wuid lines are absent. The Vrst ones are not needed for the same

reasons stated in Section 4.9.1; the latter have been eliminated since there is no

need to model the Helium purging phases in a steady state simulation. The steady

state model has been used in OU-Design mode.

The chosen input data are collected in Table 4.13; Table 4.14 summarizes the

main system variables results performed by the transient and the steady state

models. The steady state model matches very well the transient one for Wuid Wow,

turbomachinery and gas generator main parameters.

100


Name Description Value UnitsPin,ox Total pressure in LOX tank 2.0 [bar]Pin,fu Total pressure in LH2 tank 3.0 [bar]Pout,ox Total pressure at Pump outlet/Gas Generator inlet 50.0 [bar]Pout,fu Total pressure at Pump outlet/Gas Generator inlet 55.0 [bar]Pcc Initial chamber pressure 20.0 [bar]Tcc Initial chamber temperature 900 [K]ωp,ox Initial LOX pump speed 1000 [rpm]ωp,fu Initial LH2 pump speed 6000 [rpm]

Table 4.13.: HM7B Turbopump input [44] and initial data

Name Nominal Value [44] Error Transient/Steady Statemgg,ox [kg/s] 0.3%mgg,fu [kg/s] 0.07%MR [-] 0.5%Pgg [bar] 1.3%Tgg [K] 0.08%mt [kg/s] 0.3%ωt [rpm] 60500 0.8%τt [N·m] 59.98 2.0%

mp,ox [kg/s] 12.4 0.0%∆Pox [bar] 48 1.6%ωp,ox [rpm] 13000 0.8%τp,ox [N·m] 10.4%mp,fu [kg/s] 2.4 0.0%∆Pfu [bar] 52 0.8%ωp,fu [rpm] 60500 0.8%τp,fu [N·m] 6.0%

Table 4.14.: HM7B Turbopump output data

101


Figure 4.5.: Turbopump test case: HM7B power pack transient schematic

Figure 4.6.: Turbopump test case: HM7B power pack steady state schematic

102


HM7B Chamber subsystem

This test case represents the combustion chamber subsystem of the HM7B engine.

The aim of this test case is to validate the behaviour of the combustion chamber

and cooling jacket components when they are coupled together in a simulation,

by comparing results with the transient model simulation.

Figures 4.7 and 4.8 show the schematics of the combustion chamber subsystem

using the ESPSS transient library and the steady state model, respectively.

As in the previous test case, described in Section 4.9.2, the similarity of the two

schematics shown hereafter is evident. The only diUerence for the steady state

model is the absence of non-condensable Wuid lines and capacitive components

such as volumes. Also here the steady state model has been used in OU-Design

mode.

Figure 4.7.: Chamber test case: HM7B Combustion Chamber transient schematic

In Table 4.15 the main input data for both systems are collected; in Table 4.16

the main system variables results are summarized, performed by the transient and

the steady state models. As reported in the table the steady state model matches

the transient results, showing very good agreement between the values of the

combustion chamber, and of the cooling channel model.

103


Figure 4.8.: Chamber test case: HM7B Combustion Chamber steady stateschematic

Name Description Value UnitsPin,LOX Total pressure at pump outlet/chamber inlet 50.0 [bar]Pin,H2 Total pressure at pump outlet/chamber inlet 55.0 [bar]Pcc Nominal chamber pressure 36.6 [bar]nch Numbers of channels 128 [-]Pi,cc Initial chamber pressure 30 [bar]Ti,cc Initial chamber temperature 1000 [K]Po Initial total pressure in the channels 49 [bar]To Initial total temperature in the channels 30 [K]mo Initial mass Wow in the channels 2 [kg/s]

Table 4.15.: HM7B CC input [44] and initial data

Name Nominal Value [44] Error Transient/Steady Statemox [kg/s] 12.4 2.4%mfu [kg/s] 2.46 0.38%mtot [kg/s] 14.86 2.0%MR [-] 5.0 2.9%Pcc [bar] 36.6 0.32%Tcc [K] 0.88%

mch [kg/s] 2.46 0.37%∆Pch [bar] 9.9%Tout,ch [K] 6.7%

Table 4.16.: HM7B CC output data

104


4.9.3. Engine cycle designs

At the beginning of a design analysis, a set of performance parameters must be

chosen as assumption to deVne the engine class and the initial condition of the

engine:

- Propellants

- Tank pressure and temperatures

- Chamber Pressure Pc

- Chamber Mixture RatioMR

- Combustion eXciency ηc∗

- Throat Diameter Dt

- Pump eXciencies ηp

- Pump speciVc speeds Ns

- Turbine eXciencies ηt

Subsequently, it is possible to evaluate other important characteristics such as

the contour of the chamber (by use of L* and simple geometrical correlations) and

injector pressure drops.

HM7B rocket engine system

The HM7B is a gas generator cycle with single turbine and geared pumps. The

two subsystems modelled in the previous work have been updated to the current

library implementation and linked together to build the HM7B engine system

model.

Figure 4.9 shows the schematic of the HM7B engine using the Steady State

library. All model components are in Design mode.

In Table 4.17 the main input data for the system are collected; in Table 4.18 the

main system variables results are summarized and compared with nominal data.

Where nominal values are shown, they are taken from the open literature.

105


Figure 4.9.: HM7B engine system schematic

106


Name Description Value UnitsPin,LOX Total pressure in LOX tank 2.0 [bar]Tin,LOX Total temperature in LOX tank 91.2 [K]Pin,LH2 Total pressure in LH2 tank 3.0 [bar]Tin,LH2 Total temperature in LH2 tank 21.0 [K]Pcc Nominal chamber pressure 36.6 [bar]MR Nominal chamber mixture ratio 5. [-]nch Numbers of channels 128 [-]Ti,cc Initial chamber temperature 1000 [K]Po Initial total pressure in the channels 50 [bar]To Initial total temperature in the channels 30 [K]mo Initial mass Wow in the channels 2 [kg/s]Pc,gg Initial gas generator pressure 20 [bar]Tch Initial gas generator temperature 900 [K]

rpmox Initial LOX pump speed 1000 [rpm]Ns,ox LOX pump speciVc speed 10.95 [-]rpmfu Initial H2 pump speed 6000 [rpm]Ns,fu H2 pump speciVc speed 9.29 [-]mo,tu Initial Turbine mass Wow 0.2 [kg/s]

Table 4.17.: HM7B input [44] and initial data

Pressure drop has been Vxed in each valve and junction as well as turbomachin-

ery eXciency and gas generator mixture ratio. For the turbine, the “known_PI_tt”

type is chosen, while the pump speciVc speed Ns has been calculated from design

data.

The main chamber pressure and mixture ratio are Vxed. Propellant mass Wows

to the main chamber are calculated, and fed back to the upstream components.

The gas generator mass Wow is not Vxed. Only its mixture ratio is Vxed, in

accordance with the maximum allowable temperature in the turbine. The mass

Wow is then calculated by an algebraic equation system resulting automatically

from the connection of gas generator and turbopumps. The needed shaft power

drives the total gas generator mass Wow rate, since the turbine pressure ratio is

Vxed by design.

Initial values such as mass Wow rates and shaft speed are chosen by rough engi-

neering assessments. The robustness of the Steady State library is demonstrated

107


Name Nominal Value [44] Errormox,gg [kg/s] 0.59%mfu,gg [kg/s] 0.23%mt,gg [kg/s] 0.3%Pgg [bar] 0%Tgg [K] 1.03%mcc,ox [kg/s] 0.88%mcc,fu [kg/s] 0.83%mcc,t [kg/s] 14.86 0.87%mch [kg/s] 2.36%∆Pch [bar] 0.5%Tout,ch [K] 6.7%ωt [rpm] 60500 0.16%τt [N·m] 4.48%Wt [W] 4.32%Tin,t [K] 11.3%mp,LOX [kg/s] 0.96%∆PLOX [bar] 48 0.41%ωp,ox [rpm] 13000 0.16%τp,ox [N·m] 1.39%mp,LH2 [kg/s] 7.56%∆PLH2 [bar] 52 1.29%ωp,fu [rpm] 60500 0.16%τp,fu [N·m] 6.0%

Table 4.18.: HM7B engine system output data

108


by the stability of the simulation in a wide range of initial conditions.

From Table 4.18 a very good agreement with nominal data is recognisable. Few

parameters have an higher percentage error: the fuel mass Wow rate in the pump

does not take into account the dump and the tap-oU mass Wow rate vented from

the engine. The cooling jacket exit temperature takes into account the temperature

increase in the injector dome. If compared to the LH2 injector dome temperature,

the error decreases to 2.47%. Turbomachinery parameters show quite good results;

the diUerences are mainly due to the turbine inlet temperature that is lower then

the nominal one.

109


RL-10A-3-3A rocket engine system

The RL-10 is an expander cycle with single turbine and geared pumps. Being a

closed cycle, the cycle design is more diXcult than for an open cycle, because all

parameters are strongly dependent from each other.

In this test case a model of the RL-10A-3-3A rocket engine system has been

created and simulated in Design mode. Model results from the steady state

calculations have been compared with the typical engine performance parameters

at nominal operating conditions [15, 4].

Pressure drops of the main valves and junctions are taken from engine typical

values [15] as well as for the tank pressure and temperature conditions. Pumps and

turbine eXciency are Vxed to a constant value and taken from open literature [15,

4].

No calibration has been adopted for the control valves: the Oxidiser Control

Valve (OCV) aperture ratio has not been trimmed because the mixture ratio

is assigned in the combustion chamber. Mass Wows are given by combustion

chamber conditions. Turbomachinery power is regulated by the Thrust Control

Valve (TCV) that is open at its nominal open area ratio of 9% [58]. In Table 4.19 the

main input data for the system is collected; in Table 4.20 the main system variables

Name Description Value UnitsPin,LOX Total pressure in LOX tank 2.43 [bar]Tin,LOX Total temperature in LOX tank 97.056 [K]Pin,LH2 Total pressure in LH2 tank 1.86 [bar]Tin,LH2 Total temperature in LH2 tank 21.44 [K]Pcc Nominal chamber pressure 32.75 [bar]MR Nominal chamber mixture ratio 5.055 [-]nch Numbers of cooling channels 180 [-]Ti,cc Initial chamber temperature 1000 [K]mo Initial mass Wow in the channels 2 [kg/s]

Table 4.19.: RL-10A-3-3A input and initial data

results are summarized, performed by the steady state model and compared with

110


Figure 4.10.: Schematic of the RL-10 engine

111


performance parameters at nominal condition.

Name Nominal Value Errormcc,ox [kg/s] 13.95 0.38%mcc,fu [kg/s] 2.76 0.38%mcc,t [kg/s] 16.71 0.38%mch [kg/s] 2.76 0.38%Tout,ch [K] 213.44 0.18%ωt [rpm] 31 537 0.39%mt [kg/s] 2.69 1.1%Wt [kW] 588.36 3.85%mp,LOX [kg/s] 13.95 0.38%ωp,ox [rpm] 12 948 0.4%Wp,ox [kW] 82.026 4.07%mp,LH2 [kg/s] 2.7946 0.85%ωp,LH2 [rpm] 31 537 0.39%Wp,LH2 [kW] 501.86 3.68%Thrust [kN] 73.4 2.89%Isp [s] 444 2.5%

Table 4.20.: RL-10A-3-3A engine system output data

The chamber pressure is assigned together with the mixture ratio. The pressure

cascade for both propellant lines is calculated by the model according to design

parameters such as valve pressure drops. The pressure rise for the LOX pump

results directly from the calculated LOX pressure cascade, whereas an algebraic

loop is solved for calculating the needed H2 pump pressure rise and the turbine

pressure ratio. In parallel, several other algebraic loops are solved, determining

key variables such as cooling channel outlet temperature and turbine mass Wow.

The nominal fuel bypass Wow ratio [58] has been Vxed using the split component.

The turbine evaluates the required pressure ratio and the mass Wow is then

evaluated via the Thrust Control Valve (TCV).

The cooling channel component evaluates iteratively with the combustion

chamber component the heat Wuxes and the wall temperatures. Finally, pumps

112


power is evaluated from the required pressure rise and mass Wow, and rotation

speed from the given speciVc speed Ns.

The RL10 design model here described has proven to be very robust with respect

to initial conditions. The comparison of initial conditions in Table 4.19 and results

in Table 4.20 demonstrates this assertion. For example, the initial mass Wow in the

cooling jacket mo is 2 kg/s, whereas the simulation result yields 2.76 kg/s.

From Table 4.20 a very good agreement with nominal values is recognisable.

113

5. Transient Modelling

In this chapter we would like to introduce the new models developed for a better

assessment of the phenomena occurring in the subsystems of liquid rocket engines

during start-up. Three new, more complex and accurate models will be presented

in this chapter: the Vrst one for the injector plate, a second one for the evaluation

of the heat transfer coeXcient on the hot gas side of the thrust chamber, and the

third one for the evaluation of the thermal stratiVcation inside high aspect ratio

cooling channels (HARCC).

5.1. Injector Plate model

The injector head’s main task is to merge, mix and atomize the oxidizer coming

from the main valve with the fuel coming from the cooling channels. Figure 5.1

shows a schematic illustration of an arbitrary injector head, in order to clarify a

common structure. At Vrst both propellants enter separate volumes in which they

are uniformly distributed among the injector elements. Afterwards the injector

elements dose the amount of propellant mass Wow, by a deVned pressure drop and

atomize the propellants.

For the computer model and for the mathematical model formulations, respec-

tively, the conVguration of both propellant lines in the injector head are simpliVed

in that way, that one volume and one oriVce are assumed in each line. The vol-

ume represents a collector where the propellant is distributed among the injector

elements. Following, the oriVces represent the in- and outlet of all the injector

elements. In order to calculate the right Wow velocity and Reynolds number in

the injector element, one has to take the associated mass Wow rate into account.

Figure 5.2 (a) shows the component wise connection of the above mentioned

volumes and oriVces. Additionally, the convective and radiative heat transfer from

the combustion chamber into the injector head has to be regarded. In this process

114


Figure 5.1.: Schematic illustration of an arbitrary injector head

a heat Wux is transferred from the combustion chamber into the fuel cavity Vrst

and subsequently into the oxidizer cavity.

The original injector plate model present in the ESPSS library features a very

simpliVed thermal model. Indeed the injector plate topology takes into account

the eUects of the radiative heat transfer, but the conductive and convective heat

Wuxes are evaluated using only a virtual conductance. As in Figure 5.2 (a) the

original injector plate is built by a radiative and a conductive component linked

upstream in parallel directly to the combustor hot gases. These two components

are linked to a capacitive component, used to simulate the total thermal inertia

of the injector cavity walls. This heat capacity is then connected to the two Wuid

cavities.

The original model is here described:qcond = λc,hg (Tcore − Tcap)

qrad = σ (T 4core − T 4

cap)

qcap = qcond + qrad + qcav,ox + qcav,fu

115


(a) topology schematic of the original injector plate

(b) topology schematic of the new injector plate

Figure 5.2.: Schematics of the injector plates

116


where heat Wux of the capacitive component is

qcap = CpMdTcap

dt

and for the cavities components

qcav,ox = hc (Tw,cav − Tcav)

qcav,fu = hc (Tw,cav − Tcav)

In this way it is not possible to evaluate the presence of convective heat transfer

on injector face plate and to take into account the correct eUect of the conductive

and capacitive behaviour of the injector plate material. Moreover, the core tem-

perature in the Vrst volume of the chamber is considered as the wall temperature

of the injector plate, and this is unrealistic.

For this reason, an upgraded version of the injector plate topology inside the

ESPSS library has been implemented [35]. The aim of this new model is to take

into account the convective and radiative heat transfer between the Wuid in the

Vrst volume of the chamber and the face plate, and evaluate the conductive and

capacitive eUect of the injector walls in more accurate way, representative of

a generic injector head. The new structure of the injector plate (see Figure 5.2

(b)) wants to maintain the level of simplicity of the original model in order to

keep the computational cost low, and to be applicable to several diUerent injector

geometries (impinging, coaxial, etc. . . ), but in the same time wants also to improve

the heat transfer characteristics from the chamber to the injector cavities. In the

Vrst volume of the chamber the convective and radiative heat Wuxes to the injector

face are evaluated:

qconv = hc,hg (Taw − Tw,hg)

qrad = σ (T 4core − T 4

w,hg)

using the mass and the material properties of the injector plate (heat capacity, ther-

mal conductivity) the model evaluates the conductive heat transfer and capacity

eUect of the walls:

117


qcond|ox,fu =

(λ

tox,fu

)(Tw,hg − Tw,cav)

and for the capacitive componentsqcap,hg = qcond,ox + qcond,fu + qconv + qrad

qcap,ox = qcond,ox + qcav,ox

qcap,fu = qcond,fu + qcav,fu

qcap,k = CpMkdT

dtwith k = hg, ox, fu;

The thermal conductivity and heat capacity values are function of the chosen

material of the injector plate and of its temperature. For simplicity reasons the

injector plate is assumed to be made of only one material. The thickness t used for

the evaluation of the conductive heat Wux has to be considered as a “characteristic”

injector head thickness or width, and presents two diUerent values, one for the ox

side and the other for the fuel side. The capacitive components for each propellant

side are divided in two parts in order to obtain three diUerent temperatures: Tw,hg,

Tox,cav, Tfu,cav, respectively the temperature of the injector plate on the hot side,

the oxidizer and the fuel cavity wall temperature on the cold Wuid side.

5.1.1. Qualitative behaviour

In order to validate the behaviour of the new injector plate model, a numerical

approach has been used because no experimental results were found in open

literature. A pressure fed propulsion system has been modelled and tested with

both injector plate models. The test case represents a typical spacecraft propulsion

system supplied by nitrogen tetroxide (NTO) as oxidiser, and monomethylhy-

drazine (MMH) as fuel. The system is designed in order to reach a chamber

pressure of ≈ 10 bar with a mixture ratio of 1.65 at steady state conditions.

Figure 5.3 compares the two models by assessing the thermal behaviour inside

the injector cavities and the injector plate walls.

Table 5.1 summarizes the major features for each side of the injector plate

evaluated by the new model at steady state conditions.

It is evident that the temperature at injector plate wall in the original version of

118


Figure 5.3.: Temperature proVles from original and new model

Table 5.1.: Injector plate variables comparison

Variable Fuel Oxidiser ChamberInputPropellant MMH N2O4

Injector material Titanium TitaniumInjector head mass [kg] 1.5Injector head thickness [m] 0.001 0.003Injector head area [m2] 0.023 0.023 0.023Chamber pressure [bar] 9.85Chamber temperature [K] 3002.8Mixture ratio [−] 1.65hc coeXcient [W/m2 ·K] 450Inlet temperature [K] 300 292.3OutputInj. heat Wux [W] 17049 9835 26884Cavities ∆T [K] 4.85 3.10Wallc,hg temperature [K] 440.8 402.3 493.7

119


the model would have been unrealistic (Thg = Tc = 3002.8 K). Only the presence

of the virtual conductance allows the injector cavities not to increase the Wuid

temperature to unrealistic values.

Using the newly developed model, the software is able to deliver reasonable

outputs with physically valid geometries. Moreover it is possible to obtain diUerent

cavities wall temperatures for each propellant side, while before it could not occur.

5.2. Hot Gas side heat transfer coeXcient models

5.2.1. Models implemented

In the ESPSS library the heat transfer coeXcient inside the combustion chamber is

evaluated using the well-known Bartz correlation [9]. The original formulation of

this equation does not take into account several aspects, such as the combustion

zone due to atomization, vaporization and combustion delays in the proximity

of the injector plate, the boundary layer growth through the cylindrical part of

the chamber, the correct evaluation of the Wow acceleration in the convergent-

divergent part of the nozzle, etc. . .

The heat Wux in ESPSS takes into account the convective and radiative phenom-

ena:

qw = hcAwet(Taw − Tw) + σAwet(T4core − T 4

w) (5.1)

Since many correction factors used in literature are based on Stanton type corre-

lations, it was decided to use this kind of dimensionless number to evaluate the

heat transfer coeXcient.

St =hc

ρ∞v∞Cp,ref=

q

ρ∞v∞Cp,ref (Taw − Tw)(5.2)

the Stanton number represents the ratio between heat transferred to a Wuid and

the thermal capacity of this Wuid. In the combustion chamber model three diUerent

correlations have been implemented [35]:

• Original Bartz Equation

• ModiVed Bartz Equation

120


• Pavli Equation

In order to use the Bartz equation with Stanton type correction factors, the Bartz

equation has been rewritten as a Stanton type equation:

StBartz = 0.026

(µ0.2

ref

C0.6p,ref

)(λref

µref

)0.6

(m)−0.2A0.1

(Dthπ/4

Rcurv

)0.1

(5.3)

where the thermodynamic and transport properties are calculated at the so-called

Vlm temperature calculated as: Tref = 0.5 (Tst +Tw). To improve the behaviour of

the original Bartz equation, a temperature correction factor KT was added, taking

into account that the new reference temperature is calculated halfway between

the wall and the free stream static temperature. Moreover, since the geometric

reference parameter in the original Bartz equation was the throat diameter, a

further correction factor Kx was added for the consideration of the boundary

layer growth in the cylindrical part and in the nozzle [5]:

KT =

(Taw

Tref

)a

Kx =

(x

xth

)b

(5.4)

hence, the Stanton type modiVed Bartz equation becomes:

StBartz,mod = StBartz KT Kx (5.5)

Because of its simplicity, the Pavli equation has been implemented as well. The

Pavli equation including the two correction factors discussed before is [103], [5]:

StPavli = 0.023Re−0.2Pr−0.6

(Taw

Tref

)e(x

xth

)f

(5.6)

The Reynolds number Re is calculated with respect to the local chamber di-

ameter and the property reference temperature is an averaged boundary layer

temperature. In this equation the temperature correction factor and the streamwise

correction factor are also included.

In order to improve the heat Wux model in the combustion chamber another

correction factor was added taking into account the vaporization phenomenon

near the injector plate. In the so-called combustion zone the heat Wux decreases

when getting closer to the injector plate. This behaviour is due to the incomplete

121


mixing and reaction of the Wow for the given injector and combustion chamber.

The mixing region has a Vnite length where the combustion is less eUective,

therefore the heat Wuxes are lower. This correction factor is applied by using a

Stanton type correlation derived from Bartz or Pavli equations. The combustion

length can be given as an input or computed by a geometrical correlation. In

the latter case, to generate a speciVc correction factor two steps are required:

Vrst, the length of this combustion zone xmax has to be calculated based on the

injector plate geometry; then a functional dependency of the heat Wux in the range

x0 ≤ x ≤ xmax has to be found. Once the combustion zone length is evaluated

it is possible to calculate a correction factor by means of a tangential Stanton

number dependency [5]:

St∗(x)

St(xmax)=

1

4arctan

[7

(x

xmax− 0.63

)]+ 0.7 (5.7)

The last correction factor added to achieve a better agreement between the

numerical and the experimental results is a correction factor Kacc related to the

Wow acceleration. In fact, the measured heat Wuxes are lower than the calculated

ones upstream and downstream the nozzle throat.

The behaviour is caused probably by the nozzle contour and therewith due to

the Wow acceleration (bigger boundary layer thickness). Instead of using the local

velocity gradient to develop the correction factor, a more practicable way is to use

the absolute value of the Vrst derivative of the chamber radius with respect to the

streamwise coordinate, |dr/dx|.The correction factor Kacc requires two boundary conditions. In the cylindrical

part Kacc should be equal to 1. The other boundary condition is described by

Kacc = 0 and represents the disappearance of convective heat transfer due to Wow

separation. The following correction is used to take into account both conditions

[5]:

St∗ = St ·Kacc = St ·

√1−

∣∣∣∣drdx∣∣∣∣ (5.8)

122


5.2.2. Validation

In the years 1999-2000, in the frame of an ESA GSTP-2 contract, Astrium per-

formed a series of experiments with a water cooled calorimetric combustion

chamber [5]. The diUerent correlations aforementioned based on Bartz ([5],[9])

and Pavli ([5],[103]) have been implemented to simulate the calorimetric combus-

tion chamber tests and their results have been compared with the experimental

results from this calorimetric chamber test campaign [57, 56].

The calorimetric chamber is a sub-scale, water cooled thrust chamber with

twenty segments [116]. Each segment features an independent water feed system

with volume Wow measurement. For each segment the heat Wux is measured

individually. The described calorimetric system has been modelled using the

following components [116]:

• 1 combustion chamber with 21 nodes (component to validate)

• 20 regenerative circuits with 5 nodes each

• 20 mass Wow regulated water feed lines (with the necessary junctions and

boundary conditions)

• mass Wow regulated propellant feed lines

The simulation was performed using the couple liquid oxygen/gaseous hydrogen

as propellants, at an O/F ratio varying from 5 to 7 and at a total pressure from 35

to 70 bar in the combustor; for each test point the propellant mass Wows are chosen

in order to get the desired pressure and mixture ratio. In order to get the right

pressure drop through the cooling circuit, the roughness of the cooling channels

had to be adapted. Values between 3.2 and 25 µm were chosen. This tuning was

necessary because of the partially unknown layout of the cooling circuit and its

feed lines (pipes, Vttings, . . . ).

The heat Wuxes calculated with every correlation described in the previous

chapter are plotted in Figure 5.4 (a) for the nominal case (p = 60 bar, MR = 6);

in Figure 5.4 (b) simulation results are compared to experimental data obtained

varying the chamber pressure (p = 35, 60, 70 bar,MR = 6), Figure 5.4 (c) shows

the simulation and experimental comparison for tests with constant chamber

123


pressure but diUerent mixture ratio (p = 60 bar, MR = 5, 7), while in Figure 5.4

(d) the hot gas wall temperature trend is shown at diUerent chamber pressures.

The correlations are all plotted with lines and experimental data with symbols.

(a) Heat Wuxes at MR = 6, pc = 60 bar (b) Heat Wuxes at MR = 6, pc = 35, 60, 70 bar

(c) Heat Wuxes at MR = 5, 7, pc = 60 bar (d) Wall temperatures at MR = 6, pc =35, 60, 70 bar

Figure 5.4.: Heat Wuxes and wall temperatures results

The “Combustion zone” correction factor is able to represent the lower heat

Wuxes in the Vrst part of the chamber. Unfortunately, the introduced correction

cannot be considered predictive (that is, a correction that would give good results

124


in a diUerent combustion chamber and injector face): it would require experimen-

tal data with diUerent calorimetric chambers and diUerent injector conVgurations.

This is out of the scope of a 0-D/1-D investigation.

The heat Wuxes in the divergent part are always overpredicted. This is a

characteristic of the Bartz model and needs to be kept in mind when interpreting

the results. However, introducing the “Wow acceleration” correction factor is

possible to achieve a better agreement with the experimental results.

Moreover, unlike the combustion zone correction factor, its behaviour is not

peculiar of the experiment considered so it can be used for diUerent chamber

conVgurations and performance conditions. No tuning has been performed on

the Bartz and Pavli parameters, the constants have been taken as C = 0.026, and

C = 0.023 respectively as recommended by Bartz and Pavli.

For each correlation, some remarks follow:

• Simple Bartz correlation. Here, the heat Wuxes are underpredicted (around

30% in the cylindrical part) and the decreasing heat Wux in the cylindrical

part is not shown, but the shape of the curve in the convergent divergent

nozzle region is similar to the experimental one.

• ModiVed Bartz correlation. Here, the heat Wuxes are slightly overpre-

dicted, but using the temperature and the streamwise correction it has the

advantage of following very accurately the experimental data in the cylindri-

cal part. Therefore, the model without a “combustion zone” correction factor

can be applied only to part of the combustion chamber, after the mixing has

taken place.

• Pavli correlation. This correlation is able to follow the experimental trend

but in a diUerent way of the ModiVed Bartz correlation. In fact, the Pavli

correlation underestimates the heat Wuxes while the Bartz correlation over-

estimates them.

• Pressure dependency. Using the modiVed Bartz correlation, test cases at

diUerent pressures have been modelled in EcosimPro. The results shown in

Figure 5.4 (b) indicate a very good agreement with experimental heat Wux

125


values. Therefore, this correlation can be considered reliable for LOX/H2

combustion atMR = 6.

• Mixture ratio dependency. The same approach has been taken for the

mixture ratio dependency. Test cases atMR varying from 5 to 7 have been

modelled in the code. As can be seen in Figure 5.4 (c), the results present

a diverging behaviour. In particular, an increase in MR yields a general

increase in experimental heat Wuxes, while the modiVed Bartz correlation

shows the opposite trend. It is diXcult to indicate a clear explanation for

these results. The main drivers for the convective heat Wuxes are the mixture

heat capacity at the reference temperature Cp,ref and the temperature gradi-

ent (Taw−Tw). WhenMR increases, the heat capacity decreases (because of

less hydrogen in the mixture), whereas the temperature gradient increases.

In the modiVed Bartz model, it seems that of these two counteracting prop-

erties, the variation in Cp,ref is predominant. In the experiment, localMR

variations at the wall might be responsible for the opposed trend.

5.3. Q-2D stratiVcation model for HARCC

For new engines the use of High Aspect Ratio Cooling Channels (HARCC) is

necessary. Indeed, the use of these kinds of channels permits a lower wall

temperature and a longer life. Beside these advantages, the HARCC have as

usual also drawbacks: the pressure drop is higher and thermal stratiVcation

occurs within them. In order to optimize the design of this kind of channels it is

fundamental to evaluate the thermal stratiVcation eUect and so the heat absorption

of the coolant. It is therefore necessary to reVne the models developed in the

system modelling tools in order to obtain more capabilities, using speciVc models

for each cooling system adopted.

5.3.1. Model description

As compared to two diUerent papers from the Department of Mechanics and

Aerospace Engineering (DIMA) of “Sapienza” University of Rome [106] and the

German Aerospace Center (DLR) [150] that found their own way to analyse the

126


HARCC, a new approach [35] is here proposed to evaluate thermal stratiVcation

in system tools such as EcosimPro. Starting from the one-dimensional governing

equations present in the ESPSS library:

∂u

∂t+∂f(u)

∂x= S(u) (5.9)

where

u = A

ρ

ρxnc

ρv

ρE

; f(u) = A

ρv

ρvxnc

ρv2 + P

ρvH

; (5.10)

S(u) =

−ρAkwall(∂P/∂t)

−ρxncAkwall(∂P/∂t)

−0.5(dξ/dx)ρ v|v|A+ ρgA+ P (dA/dx)

qw(dAwet/dx) + ρgvA

(5.11)

The new code presents an unsteady Q-2D model and can be considered as an

evolution of the two inspiring works presented by DIMA and DLR. The control

volumes are divided in slices, one on top of the other linked together longitudinally

by the momentum and energy viscous Wuxes. The mass conservation equation

is written in a one-dimensional form but it is calculated for each slice, while the

momentum and energy conservation equations are written in a quasi-2D form

taking into account friction, longitudinal viscous transport, wall heat Wux and

longitudinal Wuid heat Wux respectively.

Equations (5.9) and (5.10) have been modiVed in the following way, to obtain

inside each channel several longitudinal Wuid veins one on top of the other and

linked by the momentum and energy viscous Wuxes:

∂u

∂t+∂f(u)

∂x+∂g(u)

∂y= S(u) (5.12)

where

127


g(u) = Awet

0

0

τxy

qc

; τxy = µt∂v

∂y; qc = λt

∂T

∂y(5.13)

The turbulent conductivity coeXcient λt is evaluated using the empirical cor-

relation of Kacynski [66]. By the use of a constant turbulent Prandtl number we

obtain the turbulent viscosity.

λt

λ= 0.008Re0.9 Prt = 0.9 µt =

Prt λt

cp(5.14)

Hence each slice has his own velocity, and no empirical correlations are used

to evaluate the velocity proVle being automatically related to the viscous Wuxes

and the longitudinal heat Wux. To accurately describe the wall heat Wux also the

wall temperature is assumed to vary along the y direction. All thermodynamic

properties such as temperature, density and enthalpy depend on x, y and time.

(a) 1D Fluid Element (b) Q-2D Fluid Element integratedwith walls

Figure 5.5.: left: 1-D Wuid element and energy balance used for conventional 1-Dmethod; right: control volumes of the Q-2D approach integrated in 3Dwall elements

128


Initial and boundary conditions of the cooling channel are the typical ones for

capacitive components: a capacitive component receives the Wow variables (mass

and enthalpy Wows) as input in inlet and outlet and gives back the state variables

(pressure and enthalpy) as output.

The cooling channel model is built from 3 components: one quasi-2D tube and

two volumes, one at the inlet and the other one at the outlet, representing the

manifold volumes of a typical cooling jacket. Each slice is connected directly to

the volumes. The quasi-2D tube is a resistive component: it receives the state

variables as input in inlet and outlet and gives back the Wow variables as output.

Please note that no velocity proVle in the y direction has been assigned, but

each Wuid vein is aUected by viscous Wuxes and wall friction. Moreover a real

time-dependent integration has been performed, in order to evaluate the thermal

stratiVcation through the time for unsteady analysis.

The evaluation of the friction factor of each cell has been done by using a

peculiar hydraulic diameter deVned as function of the wet channel surface and

the perimeter of each volume:

Dh,i =4Ai

Pwet,i

To our knowledge it is the Vrst time that a quasi-2D approach is implemented for

pipe Wows in a system tool for transient analysis; with this model we are able to

evaluate not only the stratiVcation eUect but also the time that the coolant needs

to show this stratiVcation during the transient phase of the engine ignition.

3-D cooling channels walls

The “3D wall” components used to simulate the walls are part of the original

ESPSS library [42]. They will calculate the heat conduction in every direction

including the axial direction. This thermal component features thermal ports in

radial and in azimuth directions allowing an exact calculation of heat conduction

through the channel corners. The model has been modiVed in order to allow the

connection between its thermal ports and the quasi-2D channel ports. The walls

are divided in 5 diUerent 3-D components as shown in Figure 5.6. Each component

has a 3-dimensional discretisation in tangential, radial and longitudinal direction

129


Figure 5.6.: Cooling jacket wall mesh

(dx, dy, dz), respectively.

The formulation for this component is the typical one for conduction elements;

the thermal capacitance for each volume is deVned as:

Ci,j,k = ρCp(i,j,k) dx dy dz (5.15)

the internal heat Wows are evaluated by:

qx(i,j,k) = ki,j,k dy dz (Ti−1,j,k − Ti,j,k)/dx (5.16)

qy(i,j,k) = ki,j,k dx dz (Ti,j−1,k − Ti,j,k)/dy (5.17)

qz(i,j,k) = ki,j,k dx dy (Ti,j,k−1 − Ti,j,k)/dz (5.18)

while the energy equation is:

Ci,j,kdTi,j,k

dt= qx(i,j,k)−qx(i+1,j,k)+qy(i,j,k)−qy(i,j+1,k)+qz(i,j,k)−qz(i,j,k+1) (5.19)

As shown in Figure 5.6 only half channel has been considered because of

symmetry reasons, with left and right sides adiabatic:

130


qout,right_r = 0 (5.20)

qout,int_l = 0 qout,int_right_r = 0 (5.21)

qout,ext_l = 0 qout,ext_right_r = 0 (5.22)

5.3.2. Numerical validation

The Q-2D model for cooling channels has been validated by comparison with a

numerical test case performed by DIMA [107, 108] of a turbulent Wow of methane

in a straight channel with asymmetric heating. These calculations have been

compared with ESPSS 1D calculations and with the new Q-2D model object of this

validation. The channel is smaller than the ones used in actual rocket channels.

Indeed, the geometric and the boundary conditions have been chosen by DIMA to

obtain small values of the Reynolds number, because the computational grid size

of the 3D CFD code is function of this parameter [109].

In order to validate the correct behaviour of the new transient model, two

diUerent aspect ratios of the channel have been investigated, a Vrst channel with

aspect ratio 1 and a second one with aspect ratio 8. The length and the cross

section area of the channel have been kept the same among the two diUerent

channels. Both channels are 27 mm long and have a cross section of 0.08 mm2.

The boundary conditions are the same for both channels and for all models:

a stagnation inlet temperature of 220 K, a stagnation inlet pressure of 90 bar, a

constant temperature of 600 and 220 K at the bottom and at the top of the walls,

respectively. Along the lateral side of the channel a linear temperature distribution

is applied from 600 to 220 K.

At the inlet of the channel a pressure source provided the inlet pressure, while

at the outlet a mass Wow controlled component forced the mass Wow rate. The

outlet pressure is an output of the model. Three diUerent temperature sources

provided the correct temperature distribution for the bottom side, the lateral side

and top side of the channel respectively. To ensure a correct trend of thermal

sources during the transient phase, a conductive and a capacitive component have

been linked between each temperature source and the thermal ports of the channel.

The same conVguration has been applied for the aspect ratio 1 and the aspect ratio

131


8 channel.

Results

Figures 5.7 (c,d) show the bulk evolution of the pressure and temperature along

the channel for the aspect ratio 1 case, while Figures 5.7 (e,f) show the pressure

and temperature evolution for the aspect ratio 8 channel.

When the stratiVcation eUect is not so evident, as in the aspect ratio 1 case, the

1D model and Q-2D model have a similar trend; but when stratiVcation occurs,

as in the aspect ratio 8 channel, the diUerences among 1D and Q-2D model are

evident, and the Q-2D results are closer to the 3D-CFD ones.

Figure 5.7 (a,b) compares the cross-section temperature contours at the channel

outlet, for each studied model and for both aspect ratios discussed here. The

AR = 1 case features some temperature stratiVcation in the 3D simulation. This

has not been observed with the Q-2D model described in Section 5.3, which shows

virtually no stratiVcation. On the other hand, for AR = 8, where a consistent

stratiVcation is expected, a very good agreement can be observed between the 3D

simulations and the new Q-2D model.

5.3.3. Experimental validation

The DLR Lampoldshausen test bench features a cylindrical combustion chamber

segment with four diUerent cooling channel geometries used to investigate thermal

stratiVcation [151, 132]. Its test results have been used to validate our Q-2D model

for high aspect ratio cooling channels. The combustion chamber was designed at

DLR institute of Space Propulsion particularly for studies with interchangeable

segments.

The combustor has a combustion chamber internal diameter of 80 mm and a

nozzle throat diameter of 50 mm. Liquid hydrogen is supplied to the combustor at

temperatures as low as 50-60 K while supply pressures are in the range of 200-250

bar. The HARCC segment is a single cylindrical segment with a diameter of 80

mm and 209 mm length. The test segment has on its circumference four diUerent

cooling channel geometries, in each 90 sector the cooling ducts have a diUerent

aspect ratio.

132


(a) Temperature stratiVcation, AR = 1 (b) Temperature stratiVcation, AR = 8

(c) Bulk temperature, AR = 1 (d) Bulk pressure, AR = 1

(e) Bulk temperature, AR = 8 (f) Bulk pressure, AR = 8

Figure 5.7.: Methane bulk variables evolution along channel axis

133


Figure 5.8.: Design of the 4 sector HARCC segment

section height [mm] width [mm] channels number AR

3 9.0 0.3 152 304 4.6 0.5 136 9.2

Table 5.2.: Cooling channels geometries

Figure 5.8 shows the construction of the HARCC-segment with diUerent cooling

channel geometries. The experiment was performed using the couple liquid

oxygen/gaseous hydrogen as propellants, and liquid hydrogen as coolant.

Two pressure conVgurations have been simulated: the Vrst with a chamber

pressure of about 88 bar, and the second with a chamber pressure of 58 bar. Two

sectors have been investigated: Quadrant 4, with channel aspect ratio 9.2 and

Quadrant 3 with channel aspect ratio 30.

The geometry of the investigated cooling channels as well as the number of

channels referred to circumference of the chamber are given in Table 5.2.

For each Quadrant, four sets of thermocouples have been positioned along

the channels. In each group, 5 thermocouples have been arranged with diUerent

134


Position 1 2 3 4Distance from leading edge of the segment [mm] 52 85 119 152

Thermocouple TE1 TE2 TE3 TE4 TE5

Distance from the hot gas wall [mm] 0.7 1.1 1.5 1.9 7.5

Table 5.3.: Positioning of themocouples

distances from the hot gas side wall. Location and distance from the wall of the

thermocouples are summarised in Table 5.3.

Such temperature measurements at diUerent locations provide important infor-

mation regarding the development of stratiVcation along the channels.

Modelling

The test bench has been modelled using EcosimPro [36]. As shown in Figure 5.9,

two mass Wow sources provide the correct mass Wow rate of oxygen and hydrogen

to the combustion chamber component. Because the HARCC test segment repre-

sents only a portion of the cylindrical part of the combustor, the Vrst segment has

been modelled as adiabatic.

Thermal demux components connect the combustion chamber to the HARCC

segment. At the inlet of the channel a pressure source provided the inlet pressure,

while at the outlet a mass Wow source forced the mass Wow rate. The channel

walls features three diUerent materials: the inner side and the Vns are in copper

alloy; the external wall is built with another copper alloy and a jacket in Nickel

alloy.

Results

Figures 5.10 and 5.11 refer to Quadrant 4 with channel aspect ratio of 9.2 and

show the simulation results for the pc = 88 bar test and the pc = 58 bar test,

respectively.

Figures 5.12 and 5.13 refer to Quadrant 3 with channel aspect ratio of 30 instead

and show the simulation results for the pc = 88 bar test and the pc = 58 bar test,

respectively.

Figures 5.10 (a,b) compare the wall temperatures in the cooling channels and

135


Figure 5.9.: Schematic of the experimental test case

136


the Wuid temperatures, respectively, obtained by 1D and Q-2D simulations for the

high pressure test case, while Figures 5.10 (c,d,e,f) show the temperature values

at thermocouples positions, comparing experimental values with Q-2D and 1D

simulation results. Figure 5.11 shows the same variables for the low pressure test

case.

Hydrogen enters the channels in supercritical conditions. The hot gas side heat

transfer correlation described by Eq. (6.14) was slightly adapted to the calculated

experimental hot gas side heat Wuxes. In Equation 6.14, an adapted value of 0.0263

was taken. Hence representative hot gas conditions have been modelled in terms

of heat transfer coeXcient and combustion chamber temperatures.

From Figures 5.10 (a,b) it is evident that the behaviour of the 1D model is

completely diUerent from the Q-2D model. The Q-2D model is able to obtain

a more representative temperature trend in the radial and in the longitudinal

direction. From the contour plot it is clear that the 1D model provides a very

homogeneous temperature proVle also in the walls because it is not able to take

into account the occurring of thermal stratiVcation. The validity and the usefulness

of the Q-2D model is enhanced by the comparisons shown in Figures 5.10 (c,d,e,f):

when high aspect ratio is used, 1D models are not adequate any more.

Figures 5.12 and 5.13 show the same variables for channel aspect ratio 30. In

these Vgures the diUerence between the Q-2D and 1D behaviour compared to

experimental data is once more evident.

Looking at Figures 5.12 (c,d,e,f) the maximum percentage error obtained by the

Q-2D model, when compared to the experimental data of the Vrst thermocouple,

does not exceed 10%, while the percentage error for the 1D model is around 40%.

Better results we obtain if we compare the percentage error of the same variable

in the 58 bar test case, where Q-2D model error does not exceed 5% and 1D model

error is around 39%.

137


(a) Wall temperatures, AR = 9.2, pc = 88 bar (b) Fluid temperatures, AR = 9.2, pc = 88 bar

(c) Thermocouples temperatures, x = 52 mm,pc = 88 bar

(d) Thermocouples temperatures, x = 85 mm,pc = 88 bar

(e) Thermocouples temperatures, x = 119 mm,pc = 88 bar

(f) Thermocouples temperatures, x = 152 mm,pc = 88 bar

Figure 5.10.: Wall and Wuid thermal stratiVcation,AR = 9.2, pc = 88 bar

138


(a) Wall temperatures, AR = 9.2, pc = 58 bar (b) Fluid temperatures, AR = 9.2, pc = 58 bar

(c) Thermocouples temperatures,x = 52 mm,pc = 58 bar


(e) Thermocouples temperatures,x = 119 mm,pc = 58 bar


Figure 5.11.: Wall and Wuid thermal stratiVcation, AR = 9.2, pc = 58 bar

139


(a) Wall temperatures, AR = 30, pc = 88 bar (b) Fluid temperatures, AR = 30, pc = 88 bar

(c) Thermocouples temperatures,x = 52 mm,pc = 88 bar


(e) Thermocouples temperatures,x = 119 mm,pc = 88 bar


Figure 5.12.: Wall and Wuid thermal stratiVcation, AR = 30, pc = 88 bar

140


(a) Wall temperatures, AR = 30, pc = 58 bar (b) Fluid temperatures, AR = 30, pc = 58 bar

(c) Thermocouples temperatures,x = 52.5 mm,pc = 58 bar

(d) Thermocouples temperatures, x = 85.8 mm,pc = 58 bar

(e) Thermocouples temperatures,x = 119.1 mm,pc = 58 bar

(f) Thermocouples temperatures,x = 152.5 mm, pc = 58 bar

Figure 5.13.: Wall and Wuid thermal stratiVcation, AR = 30, pc = 58 bar

141

6. Integrated Validation: RL-10 design and

analysis

The RL-10 engine is based on an expander cycle, in which the fuel (H2) is used

to cool the main combustion chamber and the thermal energy added to the fuel

drives the turbopumps. The RL-10 rocket engine is an important component of

the American space infrastructure. Two RL-10 engines form the main propulsion

system for the Centaur upper stage vehicle, which boosts commercial, scientiVc

and military payloads from a high altitude into Earth orbit. The RL-10A-3-3A

developed by Pratt & Whitney under contract to NASA, incorporates component

improvements with respect to the initial RL-10A-1 engine.

A cryogenic expander cycle engine involves a strong coupling between the dif-

ferent subsystems. This coupling is even stronger during the start and shut-down

transients, when non-linear interactions between subsystems play a major role.

In addition complex phenomena such as combustion, heat transfer, turbopump

operation, phase change, valve maneuverings are concerned, as well as important

changes in the thermodynamic properties of the Wuids involved. A transient

model helps to reduce the number of engine tests by allowing to perform a certain

amount of parametric studies in advance of the test campaign, and thus plays an

important role in the cost and risk reduction.

The RL-10 engine has been used extensively as object of simulations in the past

years [15, 14, 13, 59, 58]. In this chapter we want to show the improvements made

in terms of modelling with respect to the other tools; indeed, the model presented

here features a 1-D discretisation not only in the cooling jacket model, but also

for most of the other components, such as the combustion chamber, the Venturi

duct and the other pipes.

In previous works [15], the combustion chamber has been modelled as a built-in

set of hydrogen/oxygen combustion tables. Here, a fully 1-D discretised chamber

142

6. Integrated Validation: RL-10 design and analysis

and nozzle features a chemical equilibrium model based on Gibbs energy mini-

mization for each section along the chamber. The present model also contains the

injector plate model described in chapter 5.1 representative not only of the capaci-

tive eUect of the injector dome mass but also of the convective and radiative heat

Wuxes from the chamber to the injector and of the conductive heat Wux between

the fuel and oxidiser injector domes. The thermal model used for the cooling

jacket component is modelled as a “real” one and a half counterWow cooling jacket.

Finally it is important to mention that, to the best of the author’s knowledge,

chill down and pre-start procedures were never simulated before with transient

system tools for the whole engine. In the present work, the cool-down (pre-

start) procedure has been simulated in order to obtain a accurate and complete

engine state at start signal (t=0). The pre-start simulation results are in very good

agreement with the few experimental data available.

6.1. Overview of the RL-10A-3-3A rocket engine

The RL10A-3-3A includes seven engine valves as shown in Figure 6.2. The

propellant Wows to the engine can be shut oU using the Fuel Inlet Valve (FINV)

and the Oxidizer Inlet Valve (OINV). The fuel Wow into the combustion chamber

can be stopped by the Fuel Shut-oU Valve (FSOV) located just upstream of the

injector plenum. The FSOV is a helium operated, two position, normally closed,

bullet-type annular gate valve. The valve serves to prevent fuel Wow into the

combustion chamber during the cool-down period and provide a rapid cut-oU of

fuel Wow during engine shut-down [110].

The fuel interstage and discharge cool-down valves (FCV-1 and FCV-2) are

pressure-operated, normally open sleeve valves. The purposes of these valves are

the following [110]:

• allow overboard venting of the coolant for fuel pump cool-down during

engine pre-chill and pre-start

• provide Vrst stage fuel pump bleed control during the engine start transient

(for the FCV-1)

• provide fuel system pressure relief during engine shut-down

143


The Thrust Control Valve (TCV) is used to control thrust overshoot at start

and maintain constant chamber pressure during steady-state operation. TCV is a

normally closed, servo-operated, closed-loop, variable position bypass valve used

to control engine thrust by regulation of turbine power. As combustion chamber

pressure deviates from the desired value, action of the control allows the turbine

bypass valve to vary the fuel Wow through the turbine [110].

The Oxidizer Control Valve (OCV) has two oriVces: one regulates the main

oxidizer Wow (OCV-1) and the other controls the bleed Wow required during engine

start (OCV-2). The main-Wow oriVce in the OCV is actuated by the diUerential

pressure across the LOX pump. The OCV valve is a normally closed, variable

position valve. The valve controls oxidiser pump cool-down Wow during the

engine pre-start cycle and during engine start transient [110].

The Venturi upstream of the turbine is designed to help stabilize the thrust

control.

Ducts and manifolds in the RL10 are generally made out of stainless steel and

are not insulated.

The combustion chamber and nozzle walls are composed of cooling tubes. A

silver throat is cast in place for the RL10A-3-3A and increases the expansion ratio

for higher speciVc impulse. The inject has 216 coaxial elements; the oxidiser is

located in the center of each element and hydrogen through the annulus. One-

hundred-sixty-two of the LOX injector elements have ribbon Wow-swirlers that

provide enhanced combustion stability.

The regenerative cooling jacket serves several functions in the RL10 engine.

The basic conVguration is a pass-and-a-half stainless-steel tubular design. Fuel

enters the jacket via a manifold located just below the nozzle throat. A set of 180

“short” tubes carry coolant to the end of the nozzle. At the nozzle exit plane, a

turn-around manifolds directs the Wow back through a set of 180 “long” tubes. The

long tubes are interspersed with the short tube in the nozzle section and comprise

the chamber cooling jacket above the inlet manifold. Coolant Wow exits through a

manifold at the top of the chamber. The cooling tubes are brazed together and act

as the inner wall of the combustion chamber and nozzle.

The fuel pump consists of two stages, separated by an interstage duct, which is

vented via the interstage cool-down valve (FCV-1) during start. Both fuel pump

144


stages have centrifugal impellers, vaneless diUusers and conical exit volutes; the

Vrst stage also has an inducer.

The LOX pump consists of an inducer and a single centrifugal impeller, followed

by a vaneless diUuser and conical exit volute. The LOX pump is driven by the

fuel turbine through the gear train. The turbopump speed sensor is located on the

LOX pump shaft [111].

The RL-10 turbine is a two stage axial-Wow, partial admission, impulse turbine.

Downstream of the turbine blade rows, exit guide vanes reduce swirling of the

discharged Wuid. The turbine is driven by hydrogen and powers both fuel and

oxidiser pumps.

There are a number of shaft seals which permit leakage from the pump discharge

in order to cool the bearings. The fuel pump and the turbine are on a common

shaft; power is transferred to the LOX pump through a series of gears. The seals,

bearings, gear train all contribute to rotordynamic drag on the turbopump.

Figure 6.1.: RL-10A-3-3A engine schematic [115]

145


Name Value UnitsFuel Turbopump1st stage impeller diameter 179.6 [mm]1st stage exit blade height 5.8 [mm]2nd stage impeller diameter 179.6 [mm]2nd stage exit blade height 5.588 [mm]Oxidiser TurbopumpImpeller diameter 106.7 [mm]Exit blade height 6.376 [mm]TurbineMean line diameter 149.86 [mm]Mass moment of inertia 0.008767 [kg·m2]Ducts & ValvesFINV Wow area 0.0041 [m2]FCV-1 Wow area 0.00038 [m2]FCV-2 Wow area 0.00019 [m2]Pump discharge duct 0.0011 [m2]Venturi (inlet - throat) 0.0023 - 0.00067 [m2]TCV Wow area a 1.01E−5 b [m2]Turbine discharge housing (inlet - exit) 0.013 - 0.003 [m2]Turbine discharge duct 0.003 [m2]FSOV Wow area 0.0021 [m2]OINV Wow area 0.0031 [m2]OCV Wow area a 3.96E−4 b [m2]Cooling jacketNumber of short tubes 180 [-]Number of long tubes 180 [-]Channel width at throat 2.286 [mm]Channel height at throat 3.556 [mm]Total coolant volume 0.0158 [m3]Typical hot wall thickness 0.3302 [mm]HGS eUective surface area 4.645 [m2]Thrust chamberChamber diameter 0.1303 [m]Throat diameter 0.0627 [m]Nozzle area ratio 61 [-]Chamber/nozzle length 1.476 [m]Number of injectors 216 [-]Injector assembly weight 6.72 [kg]

Table 6.1.: RL-10A-3-3A construction data [15]a values at nominal full-thrust conditionb this Wow area includes the discharge coeXcient for the oriVce, which is unknown

146


Figure 6.2.: RL-10A-3-3A engine diagram

6.2. Design procedure

The development of the RL-10 engine transient model has been conducted with

EcosimPro and the ESPSS library, in the upgraded version including all the

relevant models developed and described in Chapter 5.

6.2.1. Turbomachinery modelling

Pumps

The pump model makes use of performance maps for head and resistive torque.

The pump curves are introduced by means of Vxed 1-D data tables deVned as

functions of a dimensionless variable θ that preserves homologous relationships

in all zones of operation. θ parameter is deVned as follows:

θ = π + arctan(ν/n) (6.1)

147


where ν and n are the reduced Wow and the reduced speed respectively:

ν =Q

QR=min/ρin

QRn =

30ω /π

rpmR(6.2)

The dimensionless characteristics (head and torque) are deVned as follows:

h =TDH /TDHR

n2 + ν2β =

τ / τRn2 + ν2

(6.3)

this method eliminates most concerns of zero quantities producing singularities.

To simplify the comparison with generic map curves, these relations are normal-

ized using the head, torque, speed and volumetric Wow at the point of maximum

pump eXciency. These maps have been created as a combination of available test

data provided by Pratt&Whitney [15] and generic pump performance curves [25]

(see Figures 6.3, test data range in grey). Additional maps were established (not

shown here), giving a corrective factor on the pump torque, function of the rota-

tional speed ratio (also provided by P&W). The enthalpy Wow rise is a function

of the absorbed power while the evaluation of the mass Wow rate is performed

through an ODE.(m h)out = τ · ω − (m h)in

I · dmdt

=

(P +

1

2ρv2

)out

−(P +

1

2ρv2

)in

− gρin · TDH

Because of the presence of the FCV-2 valve between the Vrst and the second

stage, the fuel pump has been modelled with two separated pump components, one

for each stage. Since the oxidiser pump has only one stage, it has been modelled

with one component instead. For each pump model the main nominal parameters

have been calculated by a numerical code speciVcally developed to Vnd the

nominal value of the outlet pressure, the pump torque τp, the total dynamic head

TDH , the pump eXciency ηp and the speciVc speed Ns by use of the Pump head

and Pump eXciency curves provided by Pratt & Whitney [15]. The development

of a dedicated tool for the evaluation of the pump nominal parameters has been

necessary since there was a discrepancy between the deVnition of total dynamic

148


(a) Extended Head map for LOX and Fuelpumps

(b) Extended Torque map for LOX and Fuelpumps

Figure 6.3.: Pumps performance maps

head used in the pump model and the one used in P&W maps:

TDHtool =hout − hin

g; TDHESPSS =

Pout − Pin

ρing; (6.4)

TDHP&W =

(Pout

ρout− Pin

ρin

)/g (6.5)

The Vrst one represents the head rise given by the enthalpy diUerence between

the inlet and the outlet conditions; this deVnition has been used to match the

requested power of the pump. The second one is the head given by the pressure

diUerence between inlet and outlet and the inlet density; this is the deVnition

used in the ESPSS pump model (see Eq. 3.32). The third one is deVned using the

diUerence between the pressure on density ratio at outlet and inlet and used to

deVne the numerical value from the P&W maps.

These three deVnitions of the dynamic head can be considered the same only in

the ideal case of a pure incompressible Wuid (ρin = ρout).

As real Wuids in the pump component are used, even if the Wuid is in liquid

conditions, the density diUerence between inlet and outlet generates a discrepancy

between the aforementioned deVnitions. Moreover, using in the tool the Euler

equation of turbomachinery to calculate the power, and comparing it with Eq. 3.30

149


we obtain:

W = m∆h = m∆his

ηp= τ ω ⇒ QRρing ·

(hout − hin)is

η= τR · ω (6.6)

The term(hout − hin)is

η=

TDHR

η⇒ TDH is not the same of the one

present in “inlet mass Wow equation” (see Eq. 3.32); in fact, comparing the deVni-

tion of TDH in Eq. 3.30 with the deVnition of TDH in Eq. 3.32 we obtain:

TDHR|Eq. 3.32 =∆P

ρg6= (hout − hin) = TDHR|Eq. 3.30 (6.7)

The mismatch present in the use of two diUerent versions of the total dynamic

head could aUect the results of the simulations and the performances of pump

itself. For this reason the code developed is able to calculate a “modiVed” pump

eXciency in order to match either the pressure rise either the pump torque in the

ESPSS pump model. Since no oXcial values of the propellants leak to the gear box

were found, an iterative procedure was adopted to Vnd the correct value of the

mass Wow rate and the outlet pressure in each stage.

W , m0, Pin, Tin

Input

∆h = W/m

φ = f(m)

hout

TDHP&WηP&W

hout,is

PoutTDHcalc = TDHP&W

f(hout,is, sin)=

?

NO

NO

Pout, mhout, η

TDH

YES

Figure 6.4.: Iterative procedure for determining pump parameters

Turbine

The turbine performance maps provided by Pratt&Whitney depict the combined

performance of the two stages (see Figures 6.5 (a,b)). The Vrst one describes

the eUective area (area times discharge coeXcient) as a function of velocity

150


ratio (U/Co) for several diUerent pressure ratios. The second one describes the

combined two-stage turbine eXciency as a function of velocity ratio (U/Co) as

well.

(a) EXciency map for the Turbine (b) EUective Area map for the Turbine

Figure 6.5.: Turbine performance maps from P&W [15]

In the present study, Pratt & Whitney performance maps are transformed to

obtain the turbine performance maps used in the ESPSS turbine model. These

maps (mass Wow coeXcient and speciVc torque) are introduced by means of 2-D

input data tables as a function of velocity ratio and pressure ratio (see Figures 6.6

(a,b)):

N =r · ωCo

Π = P01 /P02 (6.8)

and the mass Wow coeXcient and speciVc torque are deVned as:

Q+ =mmap · Co

r2 P01ST =

τ

r mmapCo(6.9)

According to Eq. 6.9 and to the power balance equation τ · ω = m η∆his we

obtain the non-dimensional parameters as function of velocity ratio and pressure

151


ratio using data from the P&W maps:

τ · ω = m η(Π) ∆his ⇒ ST r mCo · ω = m η(Π) ∆his

⇒ ST C2o N = η(Π) ∆his ⇒

ST (Π, N) =η(Π) ∆his

C2o N

and for the Q+ parameter we just need to calculate the turbine mass Wow as

function of N and Π:

Aeff = CD · A = f(Π, N)

m = CD · A ·√P01 ρ01

2γ

γ − 1

[Π−

2γ − Π−

γ+1γ

]This formulation is based on the assumption that no chocking conditions occur

during the transient and at steady conditions of the turbine component.

(a) SpeciVc Torque map for the Turbine (b) Mass Flow coeXcient map for the turbine

Figure 6.6.: Turbine performance maps

6.2.2. Thrust chamber and cooling jacket modelling

The thrust chamber component, inherited from the original ESPSS library [43],

represents a non adiabatic 1-D combustion process inside a chamber for liquid

152


or gas propellants. The equilibrium combustion gases properties (molar fraction,

thermodynamic and transport properties) are calculated for each chamber volume

(node) using the minimum Gibbs energy method [55] as a function of the propel-

lant’s mixture molar fractions, inlet conditions and chamber pressure. Transient

chamber conditions (pressures, temperatures, mass Wows and heat exchanged with

the walls) are derived from 1-D transient conservation equations (refer to sec-

tion 3.4). A mixture equation between the injected propellants and the combustion

gases is applied. From the deVnition of the mixture ratio MR and derivation, the

following dynamic equation gives the MR evolution:

mox = MRmfu +d

dt(MR)

ρVc

1 +MR(6.10)

Combustion takes place when mixture ratio is within the allowed limits, the

ignition Wag is active and a minimum time (ignition delay) τ has elapsed. Mass,

energy and momentum equations are basically the same as in the pipe component

with variable cross area, Equations (6.11), (6.12).

∂u

∂t+∂f(u)

∂x= S(u) (6.11)

where

u = A

ρ

ρxnc

ρv

ρE

; f(u) = A

ρv

ρvxnc

ρv2 + P

ρvH

;

S(u) =

0

0

−0.5(dξ/dx)ρ v|v|A+ ρgA+ P (dA/dx)

qw(dAwet/dx) + ρgvA

(6.12)

The centred scheme is used to discretise the chamber, using a staggered mesh

approach (see Figure 6.7). The chamber contour has been divided in 40 volumes:

10 in the subsonic section, 10 from the throat to cooling jacket inlet manifold

153


and the last 20 volumes from there until the nozzle exit. The mesh has been

stretched and compressed in order to capture the main Wuid-dynamic phenomena

occurring along the chamber (Wuid acceleration, heat Wux in the throat region,

Mach evolution). The RL10A-3-3A has a silver throat insert that creates a sharp

Figure 6.7.: RL-10A-3-3A chamber contour [15] and discretisation

edge, not typically used and diXcult for EcosimPro to model. For this reason a

scale coeXcient factor named Rins has been added into the code; the coeXcient is

function of the silver insert geometry and the eUective throat area considering the

reduction due to viscous eUects.

The walls represented by thermal components in the Cooling Jacket component

are not included in the chamber model, but are taken as a boundary for the heat

exchange calculation instead:

qw = hcAwet(Taw − Tw) + σAwet(T4core − T 4

w) (6.13)

154


In the combustion chamber component the heat transfer coeXcient hc can

be evaluated by diUerent correlations (original Bartz equation, modiVed Bartz

equation, Pavli equation). Refer to chapter 5.2 for a detailed description of the heat

transfer correlation models. An heat transfer simulation campaign at subsystem

level has been performed in order to compare the diUerent correlations and choose

the most suitable. Then the modiVed Bartz equation has been chosen. The Bartz

equation has been rewritten in a Stanton type form and modiVed with correction

factors:

StBartz = 0.026

(µ0.2

ref

c0.6p,ref

)(λref

µref

)0.6

(m)−0.2A0.1

(πDth/4

Rcurv

)0.1

KT Kx (6.14)

The RL10A-3-3A injector plate is rather complicated, involving several diUerent

injector element designs. Most of the injector elements are co-axial, the hydrogen

in injected through annular oriVces around each LOX element. The outer con-

centric row of elements, however, inject hydrogen only (which will aUect wall

cooling). It is possible that some of the diUerences encountered in the heat transfer

model (see section 5.1) are due to not including this Vlm cooling eUect in those

predictions.

The injector plate composed by injectors and injector domes is modelled by

a component that takes into account the convective and radiative heat transfer

between the Wuid in the Vrst volume of the chamber and the face plate, and

evaluates the conductive and capacitive eUect of the injector walls in an accurate

way, representative of a generic injector head (refer to section 5.1). In order to

reWect the thermal capacity of the injector plate, the actual mass and the material

properties of the dome have been used into the model (see Table 6.1). For the

oxidiser and fuel injector oriVces, junctions components have been speciVcally

modelled to match the mass Wow and the pressure drop. Nevertheless, the geomet-

rical construction data of the injector oriVces have not been modiVed but used to

assess the pressure drop coeXcient ζ ; for each propellant injectors, considering

the oriVce area as the sum of the overall injectors, it yields:(Pcc

ρ+

1

2v2

)−(Pcav

ρ

)= −1

2ζv2 ⇒ ∆P = (1 + ζ)

m2

2ρA2

155


The cooling jacket model is constructed of 360 stainless steel tubes of type 347SS

properties. There are 180 short tubes, from inlet manifold to the turn-around one,

and other 180 long tubes, from the turn-around manifold to the injector plate. The

short and long tubes are arranged side-by-side in the nozzle section.

A new model structure has been developed and implemented just for the RL-10

cooling jacket subsystem. The model has been built with two Tube components,

the Vrst one simulating the short channels and the second describing the long

channels. The two tubes are connected together thanks to a Junction component

that models in this way the pressure drop caused by the turn-around manifold.

The component developed is able to reproduce the peculiar pattern of the cooling

channels in the nozzle section, where the long tubes are interspersed with the short

tubes. The heat coming from the chamber is then distributed to both channels

respectively.

The cooling jacket model is divided into a variable number of sections in axial

direction. Every section is made of one Wuid node of the Tube component (from

FLUID_FLOW_1D library, see Equations (6.11),(6.12)), which is simulating the

cooling channels and Vve slices of the “3D wall” components, which are simulating

the metallic walls. The walls are divided in 5 diUerent 3-D components as shown

in Figure 3.5; the contours of the actual height and width of the RL-10 channels

are shown in Figure 6.8 (a, b). Each component has a 3-dimensional discretisation

in tangential, radial and longitudinal direction (dx, dy, dz), respectively.

Since the cooling channel shape is not rectangular but slightly rounded (see

Figure 6.8 (c)), a detailed geometrical reconstruction has been performed to assess

the eUective exposed surface area, to maintain the original pressure drop and the

coolant velocity evolution. To this purpose the Pratt&Whitney speciVcation has

been accepted regarding the angle of exposure which is around 112 [15].

156


(a) Short channels width and height pro-Vle [137]

(b) Long channels width and height proVle [137]

(c) Detail of Tubular Construction

Figure 6.8.: Cooling jacket channels proVles

157


6.2.3. Lines, valves and manifolds modelling

In addition to the various subsystem listed above, there are on the RL-10 engine

a large number of lines valves and manifolds. Valves are modelled as zero di-

mensional components while the lines present in the engine are modelled via

an area-varying non-uniform mesh 1-D scheme. Where possible and data were

available a detailed geometrical reconstruction has been performed, as for the case

of the Venturi pipe and the discharge turbine pipe.

Fuel line set

The fuel line setup enables the fuel Wow from the FINV to the combustion

chamber passing through the fuel pump, the fuel discharge duct comprehensive of

a calibrated oriVce, the cooling channels, the Venturi duct, the turbine, the turbine

discharge duct and Vnally the FSOV.

The oriVce diameter present in the discharge duct is determined during the

engine calibration depending upon the discharge coeXcient value of the individual

components. The calibration oriVce is represented by a pressure drop equation

with sonic speed limitation implemented in the Junction component. The value

of the loss coeXcient is calculated to get the desired value for the pressure drop

according to Section 3.2.2.

The Venturi pipe downstream of the cooling jacket is intended primarily to

help provide stable thrust control using a turbine bypass valve rather than an

in-line valve. The RL-10 Venturi is apparently choked during engine start but not

at the normal operating conditions. The model presented by Binder [15] made

use of a performance map based on the inlet-to-exit pressure ratio. This model

was quite simple and needed the implementation of a inertial damping logic to

perform shut-down simulations. In this work a complete proVle of the Venturi

duct has been reconstructed instead, based on the data provided by Binder and

direct measurements of the component, in order to be compliant with inlet, the

throat and the diUuser exit diameters. The duct proVle has been implemented in a

pipe component which has been accordingly discretised to have a Vne mesh in

proximity of the throat.

158


Figure 6.9.: Venturi nozzle proVle

Name Value Units

Inlet Diameter 0.054 [m]

Throat Diameter 0.029 [m]

Outlet Diameter 0.054 [m]

Tube Length 0.75 [m]

Number of nodes 17 [-]

∆Pref 1.2 [bar]

mref 2.78 [kg/s]

Table 6.2.: Venturi geometrical data

For the all valves installed in the fuel line a loss coeXcient is used for the

pressure loss calculation. The determination of the loss coeXcient is described in

Equation 6.15:

ζ =2 ·∆Pref · ρref · A2

m2ref

(6.15)

The volumes associated with this device are implemented as extra volumes in

the components upstream and downstream the valve. Due to the small dimensions

of the valve, the code of the component does not need the implementation of

a heat transfer model, which is neglected. The input parameters are shown in

Table 6.3.

Oxidiser line set

The oxidiser line setup enables the oxygen Wow from the OINV to the combus-

tion chamber passing through the oxidiser pump, the OCV valve and the oxidiser

discharge duct.

Particular attention has been paid to the OCV since this component represent the

most complex valve to be modelled. As already mentioned the OCV is composed

of two oriVces and its main oriVce is actuated by diUerential pressure between

the oxidiser pump. The valve model presents two valves in parallel and they

have been calibrated in order to achieve the OCV performances during engine

start, steady state and shut-down phases. The pressure loss coeXcients have been

evaluated using the same equation as for the other valves (Eq. 6.15). The input

parameters are shown in Table 6.4.

159


Name Value UnitsFINVReference Flow Area 0.0041 [m2]ζ 1.0029 [-]∆Pref 0.0325 [bar]mref 2.8 [kg/s]τres 17 [ms]FCV-1Reference Flow Area 1.44E-6 [m2]ζ 2.7778 [-]∆Pref 36.601 [bar]mref 0.0195 [kg/s]τres 10 [ms]FCV-2Reference Flow Area 8.6E-7 [m2]ζ 2.7778 [-]∆Pref 73.835 [bar]mref 0.0163 [kg/s]τres 10 [ms]TCVReference Flow Area 1.66E-6 [m2]ζ 1 [-]∆Pref 16.030 [bar]mref 0.072 [kg/s]τres 10 [ms]FSOVReference Flow Area 0.0021 [m2]ζ 1.397 [-]∆Pref 2.634 [bar]mref 2.78 [kg/s]τres 10 [ms]

Table 6.3.: Fuel line valves parameters

160


Name Value UnitsOINVReference Flow Area 0.0041 [m2]ζ 1.054 [-]∆Pref 0.055 [bar]mref 14.207 [kg/s]τres 17 [ms]OCV-1Reference Flow Area 3.44E-4 [m2]ζ 0.88 [-]∆Pref 5.84 [bar]mref 13.006 [kg/s]τres 10 [ms]OCV-2Reference Flow Area 5.5E-5 [m2]ζ 2.685 [-]∆Pref 5.84 [bar]mref 1.201 [kg/s]

Table 6.4.: Oxidiser line valves parameters

161


6.3. Subsystem simulation: validation at nominal conditions

Each component of the RL-10A-3-3A engine has been previously simulated as a

stand-alone component to validate its behaviour at steady state conditions, then

they have been grouped in several subsystems:

- Turbopump assembly

- Thrust chamber and cooling jacket

- Oxidiser pipe line

- Fuel pump to cooling jacket pipe line

- Cooling jacket to turbine pipe line

- Turbine to chamber pipe line

All subsystem models have then been connected together to create the complete

RL-10A-3-3A engine model (see Figure 6.10. Two diUerent conVgurations of the

engine model parameters have been adopted: the Vrst one to match the engine

nominal operation point and a second one to match the ground test results. What

diUers from the two conVguration is the temperature and pressure at the inlet

of the pumps and the trimming of the OCV valve to obtain the desired Mixture

Ratio.

Nominal operation point has been considered for the steady-state performance

prediction. Flight data have not been considered in this comparison because insuf-

Vcient data exist to determine the mixture ratio and trim position of the oxidiser

control valve (OCV). Table 6.5 shows relative performance predictions of the tran-

sient model at steady state conditions. Where available, experimental values at

the end of the transient phase have been used as reference [15]; other performance

parameters have been compared at their nominal operating condition [15, 4, 58].

162


Name Description Value ErrorPcc [bar] Chamber pressure 32.696 -0.16%MR [-] Mixture Ratio 5.025 -0.58%mcc,ox [kg/s] LOX chamber mass Wow 14.102 -0.38%mcc,fu [kg/s] H2 chamber mass Wow 2.806 +0.77%mcc,t [kg/s] Total chamber mass Wow 16.908 +0.09%∆Pcj [bar] Cooling jacket pressure drop 13.877 +0.136%Tin,t [K] Turbine inlet temperature 204.235 -4.31%Πtt [-] Turbine pressure ratio 1.403 -0.33%ωt [rpm] Turbine rotational speed 31541 -0.015%τt [N·m] Turbine torque 180.47 -1.3%mt [kg/s] Turbine mass Wow 2.784 -1.35%Wt [kW] Turbine power 596.103 -1.31%mp,ox [kg/s] LOX pump mass Wow 14.102 -0.38%ωp,ox [rpm] LOX pump rotational speed 12949 -0.015%τp,ox [N·m] LOX pump torque 63.476 -5.81%Wp,ox [kW] LOX pump power 86.082 -5.8%mp1,fu [kg/s] H2 1st stage mass Wow 2.842 -0.77%mp2,fu [kg/s] H2 2nd stage mass Wow 2.822 -0.77%ωp,fu [rpm] H2 rotational speed 31541 -0.015%τp1,fu [N·m] H2 1st stage torque 73.158 +2.17%τp2,fu [N·m] H2 2nd stage torque 79.626 +0.16%Wp1,fu [W] H2 1st stage power 241.645 -0.32%Wp2,fu [W] H2 2nd stage power 263.01 -0.77%Thrust [kN] Engine thrust 72.352 -1.42%Isp [s] Engine speciVc impulse 440.751 -0.69%

Table 6.5.: RL-10A-3-3A engine system output data

163


Figure 6.10.: RL-10A-3-3A schematic model

164


6.4. RL-10 Engine start-up

6.4.1. Description of the start-up sequences

The RL-10 engine starts by using the pressure diUerence between the fuel tank

and the nozzle exit (upper atmospheric pressure), and the ambient heat stored in

the metal of the cooling jacket walls. The engine “bootstraps” to full-thrust within

two seconds after ignition.

Before engine start, FINV and OINV valves are opened and propellants are

allowed through the fuel pump for Vve seconds (cooled to prevent cavitation at

engine start) and through the LOX system for nine seconds. This “pre-start” Wow

consumes approximately 10 kg of oxygen and 2.7 kg hydrogen [115]. The fuel

FCV-1 and FCV-2 valves (see Figure 6.2) are open and the main shut-oU valve

(FSOV) is closed. The fuel Wow is vented overboard through the cool-down valves

and does not Wow through the rest of the system; the latent heat in the metal of

the combustion chamber cooling jacket is therefore available to help drive the

start transient. The oxidiser pump is pre-chilled by a Wow of oxygen, which passes

through the Oxidiser Control Valve (OCV) and is vented through the combustion

chamber and nozzle.

A typical plot of the valve positions during engine start is shown in Figure 6.11.

To initiate start, the FSOV is opened and the fuel-pump discharge cool-down

valve (FCV-2) is closed. The interstage cool-down valve (FCV-1) remains partially

open in order to avoid stalling of the fuel pump during engine acceleration. The

pressure drop between the fuel inlet and the combustion chamber drives fuel

through the cooling jacket, picking up heat from the warm metal. This pressure

diUerence also drives the heated Wuid through the turbine, starting rotation of

the pumps, which drive more propellant into the system. At start, the OCV also

closes partially, restricting the Wow of oxygen into the combustion chamber. This

is done to limit chamber pressure and ensure a forward pressure diUerence across

the fuel turbine after ignition of the thrust chamber.

Ignition of the main combustion chamber usually occurs approximately 0.3

seconds after the main-engine start signal (t = 0) is given (for Vrst-burns). The

ignition source is an electric spark, powering a torch igniter. The ignited com-

bustion chamber provides more thermal energy to drive the turbine. As the

165


Figure 6.11.: RL-10A-3-3A Valve schedule for Start-up Simulation [15]

166


turbopumps accelerate, engine pneumatic pressure is used to close the interstage

cool-down valve completely and open the OCV at pre-set fuel and LOX pump

discharge pressures. The OCV typically opens very quickly and the resultant

Wood of oxygen into the combustion chamber causes a sharp increase in system

pressures. During this period of fast pressure rise, the thrust control valve (TCV)

is opened, regulated by a pneumatic lead-lag circuit to control thrust overshoot.

The engine then settles to its normal steady-state operating point.

6.4.2. Start transient

The results of start transient simulations (“Simulation” on the plots) were com-

pared with measured data of a single ground test Vrst-burn (P2093 Run 3.02 -

Test 463, “Ground Test [3]” on the plots) [15] and with the simulation results of

a previous work (“xx_sym [3]” on the plots) performed by a NASA team [15].

Since no detailed initial conditions along the engine were available, a simulation

of the pre-start phase was necessary to obtain reasonable initial conditions for the

engine start.

The inlet pressure and temperature used in the model are coming from nominal

operation conditions. Another important variable is the cooling jacket initial

temperature, that has been set at 300 K before the pre-start phase occurs. After the

pre-start simulation the cooling jacket wall temperature decreased around 240 K.

It is clear that the cooling jacket wall temperatures have a great importance since

they help to determine the engine start capability. The cooling jacket manifold has

a lower temperature than the cooling jacket because it is partially Vlled by gaseous

hydrogen that has not vented overboard via the fuel discharge valve (FCV-2).

In the simulation the ignition occurs when the propellant mixture ratio inside

the combustion chamber reaches a value lower than 30 (as shown in Ref. [114]),

at around 0.3 seconds as in the ground test.

Figure 6.13 (a) shows the comparison between measured and predicted cham-

ber pressure. The model matches the measured time-to-accelerate to within

approximately 92 milliseconds (the “time-to-accelerate” is deVned here as the time

from 0 seconds at which the chamber pressure reaches 13.79 bar (200 psia)). The

very Vrst pressure rise at ≈0.3s represents the chamber ignition as mentioned

167


above. The chamber pressure shows a “plateau” until the OCV opens. After the

OCV opening, the chamber pressure rises very quickly and then stabilizes to the

steady state condition thanks to the TCV valve closed loop control.

The presence of small oscillations evident in the test data are due to oscillations

of the TCV servo-mechanism. Such a mechanism is absent in the model so no

oscillations occur. To obtain a reasonable chamber pressure proVle the TCV

opening sequence has been modiVed (see Figure 6.12) using the opening sequence

obtained from a dynamic model of the TCV valve as a guideline [113]. This new

opening sequence uses as a period the time that the dynamical model takes to

reach the steady condition after a sequence of oscillations. With this new opening

schedule the diUerence with the previous model of the RL-10A-3-3A transient

start-up [15] is remarkable.

Figure 6.12.: Valves opening sequence adopted in the simulation

168


In Figure 6.13 (b) the LOX pump rotational speed is shown: the simulation

result is in good agreement with experimental result. The diUerence in the rate of

change of the pump speed between the simulation and experiment may be due to

the uncertainty in the pump inertia distribution.

Figures 6.14 (a, b) depict the LOX pump inlet and outlet pressures evolution.

The simulation exhibits some sharp transient before reaching steady-state condi-

tions; these seem to be due to Wuid compressions and phase changes that occur

when the OCV suddenly opens. These transients are steeper than the measured

data probably because the dynamic behaviour of the OCV valve plays an impor-

tant role in the Wuid dynamics during pressure rise. The OCV valve component

indeed, has been modelled by an open-loop control logic, while the real component

has pressure controlled mechanism function of the inlet and outlet pump pressures.

For the same reason an oxygen mass Wow peak is present in the simulation

at the inlet of the engine, as it is illustrated in Figure 6.15 (a).

Figure 6.15 (b) shows the fuel inlet mass Wow trend; as for the chamber pressure,

also for the measured hydrogen mass Wow the evident oscillations are explained

by the oscillations of the TCV close-loop control mechanism. Unfortunately on

the fuel side no turbopump measured data are available so no comparison has

been possible between the simulated and experimental results.

The last two measured points in the engine were the pressure at the Venturi

inlet and the temperature at the Turbine inlet. The Vrst one is illustrated in Fig-

ure 6.16 (a): it is evident that the simulation evolution is in a very good agreement

with experimental data, but also here can not represent the pressure oscillations

due to the TCV valve.

The turbine inlet temperature trend is depicted in Figure 6.16 (b): the tempera-

ture value at time = 0s represents the initial condition obtained after the simulation

of the chill-down phase, explaining the diUerence from the “Simulation” line and

the other two temperature plots. The temperature proVle results very similar

between the two simulations due to the mixture ratio trend inside the chamber:

the engine keeps for most of the time a high mixture ratio condition but from time

169


= 1.5 s to 1.9 s (prior to the OCV complete opening) an increase of the hydrogen

Wow is noticed aUecting the combustion chamber temperature.

170


(a) Chamber Pressure

(b) LOX Pump Shaft Speed

Figure 6.13.: Transient results - part 1

171


(a) LOX Pump Discharge Pressure

(b) LOX Pump Inlet Pressure


172


(a) LOX Engine inlet mass Wow

(b) Fuel Engine inlet mass Wow


173


(a) Venturi inlet Pressure

(b) Turbine Inlet Temperature


174


6.5. RL-10 engine shut-down

6.5.1. Description of the shut-down sequence

The RL-10 engine switches oU at the end of its mission, after the steady state

phase. The Fuel Shut-oU Valve (FSOV) and the Fuel Inlet Valve (FINV) close as the

FCV-1 and FCV-2 valves open, allowing fuel to drain out of the system through

the overboard vents. The combustion process is soon starved of fuel and the Wame

extinguishes. The Oxidiser Control Valve (OCV) and the Oxidiser Inlet valve

(OINV) begin to close next, cutting oU the Wow of oxygen through the engine.

The turbopump decelerates due to friction losses and drag torque created by the

pumps as they evacuate the remaining propellants from the system. A typical plot

of the valve movement during engine shut-down is shown in Figure 6.17.

During the engine shut-down, a diUerent combination of oU-design conditions

appears to exist, including pump cavitation and reverse Wow. Proper simulation of

these eUects is complicated by their interaction with each other. From available

test data and simulation output, it appears that as the fuel inlet valve closes and

the cool-down valves open, the pump Vrst cavitates due to a combination of

changes in pump loading and cut-oU of the inlet Wow. The cavitation causes the

pump performance to degrade rapidly until the pump cannot prevent the reverse

Wow of Wuid as it comes backward through the cooling jacket. When the reversed

Wow reaches the closed fuel inlet valve, however, extreme transients of pressure

and Wow are created. Similar eUects are encountered in the LOX pump during

shut-down as well.

The pump head and torque performance characteristics during, this period of

operation are, of course, not extensively documented in test data. The generic

pump characteristics found in References [130] and [25] have been used again to

extend the performance maps for cavitation and reverse Wow.

The pump map extensions for engine shut-down are included in Figures 6.3,

page 149. Although the engine start-up and shut-down models use the same pump

performance maps (which should be able to cover all the pump regimes), the

cavitation and reverse Wow eUects also require additional modelling eUort, that

has not been implemented into this model yet.

175


Figure 6.17.: RL-10A-3-3A Valve schedule for Shut-down Simulation [15]

176


6.5.2. Shut-down transient

The results of start transient simulations (“Simulation” on the plots) were com-

pared with measured data of a single ground test (P2093 Run 8.01 - Test 468,

“Ground Test [3]” on the plots) [15] and with the simulation results of a previous

work (“xx_sym [3]” on the plots) performed by the NASA team [15]. DiUerently

form the start-up simulation, the uncertainty related to the valves closing sched-

ule made necessary to slightly trim the valve sequence (few milliseconds). The

original schedule and the valves positions proVle has been used as guideline. The

modiVed shut-down sequence is illustrated in Figure 6.18.

Figure 6.18.: Valves closing sequence adopted in the simulation

Figure 6.19 (a) illustrates the combustion chamber pressure trend. Once the

FSOV starts to close the chamber pressure decreases and this happens in both the

simulations and the experimental data, showing a good agreement between them.

177


Figure 6.19 (b) shows predicted and measured pump speed for the Oxidiser pro-

pellant side. The discrepancies from the two models and the ground test measured

data are imputable uncertainties to exact inlet conditions and initial operating

point as well as to a precise distribution of the turbopump assemblies inertia.

Figures 6.20 (a, b) depict the LOX pump inlet and outlet pressures evolution.

Regarding the pressure at the outlet of the pump, no special features are evident.

Once the FSOV valve starts to close, the outlet pump pressure decreases because

of the minor power delivered by the turbine.

Figure 6.20 (b) illustrates the inlet pressure instead. From the measured data we

see an initial pressure decrease due to the pump conditions and then a recovery in

the pressure to the complete closure of the OINV valve. This behaviour is barely

reproduced by the simulation because of the lack of a cavitation model in the

pump, hence the Vnal pressure decrease is not as evident as in the experiment.

The engine propellant mass Wows are depicted in Figures 6.21 (a, b), for the

oxidiser and the fuel respectively. The oxygen mass Wow behaviour (Figure 6.21

(a)) is mainly function of the pump behaviour; it is interesting to underline that

from analyses performed varying the opening/closing time of the valves, the role

of the FCV valves becomes much more evident. The opening of the FCV valves

decreases the turbine power, thus decreasing the propellant mass Wow rate in the

system in order to avoid mass Wow rate surges of oxygen at the FSOV closure.

In the end, the complete shut-oU of the OINV valve extinguishes the propellant

Wow rate.

A more complex proVle is present in the fuel Wow plot as shown in Figure 6.21

(b): at the beginning of the shut-down phase the hydrogen mass Wow at the engine

inlet increases because of the opening of the FCV valves. Then the closure of the

FSOV and of the FINV valve determine the mass Wow shut-oU. The simulation

reproduces correctly what happens at inlet of the engine, even though the amount

of mass Wow venting through the FCV valves results too high determining a

higher peak at the inlet respect to the one observed in the ground test. Another

interesting point to be mentioned is that diUerently from the NASA results the

“Simulation” line does not show any reverse Wow at the inlet as well as the experi-

178


mental data.

The RL10 shut-down model has captured many interesting eUects that occur

during shut-down. In Figure 6.22 (a), for example, the measured data show a

characteristic dip, rise and then falloU in the fuel venturi upstream pressure. This

features is caused by the dynamic interaction of the fuel pump cool-down valve

opening and main fuel shut-oU valve closing. It is very likely that the absence

of this peculiar behaviour inside our model is due to a not perfectly precise

synchronization of the fuel valves closing schedules.

In Figure 6.22 (b), the jump in pump inlet pressure is due, in part, to reverse

Wow through the fuel pump. As already mentioned, a cavitation model for pump

performance deterioration is not implemented so the pressure peak does not rise

in the simulation result.

From inspection of the plots, it appears that there are still unresolved diUerences

between the predicted and measured engine deceleration rates. The discrepancies

can be tracked down due two main causes: Vrst, the time scales of the shut-down

processes are much smaller than the one from the start-up transient, and second

the complex phenomena such as cavitation and blade to Wuid interaction that are

not taken into account into the present model.

179



(b) LOX Pump Shaft Speed

Figure 6.19.: Shut-down results - part 1

180



(b) LOX Pump Inlet Pressure



(a) LOX Engine inlet mass Wow

(b) Fuel Engine inlet mass Wow



(a) Venturi inlet Pressure

(b) Fuel Pump Inlet Pressure


183


6.6. Dynamic Response Analysis

In this section the dynamic response of the entire engine system to valve per-

turbations is investigated and illustrated. The thrust control valve TCV and the

oxidiser control valve OCV have been throttled in the range of ±10% around their

nominal aperture ratio by use of a step function.

The main objective is to understand the response of the engine when the valve

in charge of the thrust control (TCV) or the valve in charge of mixture ratio

deVnition are throttled.

The simulations performed have as initial operating condition the nominal

steady state point.

Thrust Control Valve throttling :

The thrust control Valve (TCV) has been operated in order to give an instantaneous

aperture ratio signal from 0.3 to 0.27 and from 0.3 to 0.33, that is a signal equal to

±10% of its nominal value.

When the TCV valve is operated, the main objective is to modify the thrust of

the engine. Closing or opening the bypass TCV valve we increase or decrease

respectively the fuel mass Wow rate into the turbine, varying the delivered power.

Figures 6.23 to Figures 6.26 show how the main engine subsystems react to

TCV operation, while Table 6.6 summarises the main engine parameters values at

nominal operation point, the percentage diUerence when compared to nominal

conditions, the value of τR the response time required to reach 90% of the Vnal

value at steady state conditions and some comments about the characteristics of

the curves shown hereafter (“Sym” stands for a symmetric trend of the variable

respect to ±10% of the valve opening, “Os” stands for Overshoot and “Rev” stands

for Reverse, that is when a variable presents a change in its trend).

Since the bypass valve has a very little cross section area, a 10% modiVcation on

its aperture ratio does not aUect so much the engine parameters. It is interesting

to underline that, since the TCV operation has a direct inWuence on the turbine

that control both propellants’ lines, the percentage variation with +10% or -10% is

almost the same for each parameter in both lines.

184


As expected, reducing the aperture ratio of the TCV valve we Vnd that chamber

pressure increases, mixture ratio shows a very slight increase as well as the

discharge pumps pressures (see Figures 6.24 (a,b)) and the injected propellants

into the chamber (see Figures 6.25 (a,b)). Please note that the fuel mass Wow rate

injected into the combustion chamber decreases at the very beginning instants

after the valve aperture ratio reduction; then the system starts to react to the

increase of turbine power and subsequently we see the hydrogen Wow increasing

into the chamber. The time required to the system to react is deVned as the τRparameter. From Table 6.6 it is clear that τr is between 0.3 and 0.4 s, one order

of magnitude the response time of the valve. This time is mainly function of the

inertia of the turbopump assembly and of the length of the pipes.

Another interesting point to underline is that for a modiVcation of ±10% of the

valve aperture ratio corresponds a percentage variation in the valve mass Wow

rate of almost the same quantity (see the line in Table 6.6).

185


Variable Nominal ∆-10% [%] ∆+10% [%] τR [s] NotesPcc 32.87 +0.243 -0.243 0.36 SymTcc 3252.19 +0.069 -0.068MR 5 +0.189 -0.19 0.09 Osmox 14 +0.305 -0.301 0.352 - 0.335 Symmfu 2.8 +0.116 -0.116 0.407 - 0.37 Revmcc,t 16.8 +0.273 -0.274Pinj,ox 37 +0.288 -0.286Pinj,fu 36.67 +0.224 -0.224Wt 589.99 +0.352 -0.352τt 179.22 +0.242 -0.242ωt 31437 +0.11 -0.111 SymΠtt 1.4 +0.023 -0.023Pout,ox 45.17 +0.345 -0.342 0.358 - 0.348 Symτp,ox 62.89 +0.356 -0.353ωox 12906.9 +0.111 -0.111Wp,ox 85 +0.467 -0.463Pout,fu 73.58 +0.202 -0.202 0.355 - 0.384 Symτp1,fu 72.66 +0.218 -0.218τp2,fu 79.12 +0.226 -0.227Wp1,fu 239.21 +0.329 -0.329Wp2,fu 260.47 +1.699 -1.247∆POCV 5.61 +0.607 -0.597mOCV−1 12.82 +0.304 -0.3mOCV−2 1.18 +0.305 -0.3∆PTCV 16.08 +0.293 -0.293mTCV 0.02 -9.798 +9.752

Table 6.6.: Engine dynamic response to TCV ±10% operation

186



(b) Chamber Mixture Ratio

Figure 6.23.: TCV throttle results - part 1

187



(b) Fuel Pump Discharge Pressure


188


(a) LOX chamber inlet mass Wow

(b) Fuel chamber inlet mass Wow


189


(a) Turbine Mass Flow

(b) Turbine Shaft Speed


190


Oxidiser Control Valve throttling :

The Oxidiser Control Valve (OCV) has been operated in order to give an instanta-

neous aperture ratio signal from 0.745 to 0.6705 and from 0.745 to 0.8195, that is a

signal equal to ±10% of its nominal value.

When the OCV valve is operated, the main objective is to modify the mixture

ratio of the engine. Closing or opening the control valve OCV we increase or

decrease respectively the oxidiser mass Wow rate into the oxidiser line of the

engine, varying the delivered propellant amount into the chamber.

Figures 6.27 to Figures 6.30 show how the main engine subsystems react to OCV

operation, while Table 6.7 summarises the main engine parameters values at nom-

inal operation point, the value of τR and some comments about the characteristics

of the curves shown hereafter.

Since the OCV valve operates in the oxidiser line, the behaviour of the entire

system results more complex and a deeper investigation of how system reacts is

needed. The fuel side of the engine system is involved by the valve operation

indirectly. A decreasing of the OCV aperture ratio generates a increase of the

valve resistance leading to a decrease of the oxidiser mass Wow rate as well as

an increase of the LOX pump head rise. Both contrasting phenomena lead to a

decrease of the shaft pump required torque on the oxidiser side determining an

acceleration of turbopump subsystem and involving in this way the fuel side of

the engine, in which the hydrogen mass Wow shows an increment. For these many

reasons the combustion chamber pressure shows a slight percentage increment.

The opposite behaviour is shown when we open of 10% the OCV valve. As

expected, the resistance of the valve is lower, the shaft pump torque goes up, hence

the oxygen mass Wow rate increases while the hydrogen Wow shows a decrease

because of the minor power delivered by the turbine to the pumps.

191


Variable Nominal ∆-10% ∆+10% τR [s] NotesPcc 32.87 +0.544 +0.15 0.51 RevTcc 3252.19 -1.226 +2.046MR 5 -3.533 +6.801 0.474 - 0.429 Osmox 14 -0.626 +2.385 0.53 - 0.59 Osmfu 2.8 +3.013 -4.135 0.497 - 0.47 Osmcc,t 16.8 -0.021 +1.298Pinj,ox 37 +0.331 +0.71Pinj,fu 36.67 +0.799 -0.351Wt 589.99 +4.606 -4.691τt 179.22 +3.091 -3.707ωt 31437 +1.47 -1.022 AsymΠtt 1.4 +1.048 -1.058Pout,ox 45.17 +2.647 -0.647 0.420 - 0.449 Asym-Revτp,ox 62.89 +1.503 -0.217ωox 12906.9 +1.471 -1.022Wp,ox 85 +2.995 -1.237Pout,fu 73.58 +2.12 -2.266 0.435 - 0.413 Asymτp1,fu 72.66 +2.174 -3.694τp2,fu 79.12 +4.45 -4.858Wp1,fu 239.21 +3.676 -4.678Wp2,fu 260.47 +5.986 -5.83∆POCV 5.61 +19.672 -12.06mOCV−1 12.82 -1.553 +3.178mOCV−2 1.18 +9.386 -6.202

Table 6.7.: Engine dynamic response to OCV ±10% operation

192



(b) Chamber Mixture Ratio

Figure 6.27.: OCV throttle results - part 1

193



(b) Fuel Pump Discharge Pressure


194


(a) LOX chamber inlet mass Wow

(b) Fuel chamber inlet mass Wow


195


(a) Turbine Mass Flow

(b) Turbine Shaft Speed


196

7. Conclusions

Transient phenomena in liquid rocket engines, ranging from combustion high

frequency instabilities to water hammer eUects in the feed lines, and which poten-

tially result in system failures, drive the necessity to dedicate special attention to

transient phases.

Concentrating on the behaviour of only one component is however not suXcient

to understand how components aUect each other during such phases, what is their

impact on system frequencies, and how this interaction may lead to a failure.

The simulation of the complex Wow behaviour in engine components and

components assemblies is therefore required. Models allowing the examination of

detailed component Wow behaviour are based on the equations of conservation

of mass, momentum, and energy, and vary widely in their complexity and in the

computational time each requires. An intelligent simpliVcation of the underlying

processes allows to reduce the governing partial diUerential equations to ordinary

diUerential equations, which no longer require complex solution methods thus

allowing much faster computational times.

The development of model capable of simulating in a more accurate way with

respect to previous models liquid rocket engine components and propulsion sys-

tems resulted in the work performed in this thesis. Implemented in the ESPSS

library, they can simulate the major liquid rocket engine components: pipes,

valves, injector domes, injectors, turbopumps, combustion devices and nozzles.

For the creation of a steady state library, each component has been tested to

validate its behaviour at component level and then in a more complex system. The

models developed and improved for transient analyses have been validated either

with CFD numerical test cases or experimental results. Each one of them, tested

in system, displays its own dynamics and characteristics which when integrated

in a more complex component assemblies are seen to interact.

197

7. Conclusions

A new library for steady state applications has been presented and validated.

The library enables to perform in a fast and reliable way design and parametric

analyses of liquid propulsion systems. The present work has described a complete

set of components able to perform dimensioning design studies and oU-design

analyses of liquid propulsion systems. A gas generator and an expander cycle

have been chosen to validate the design capability of the steady state library.

The resulting designs have been compared with actual liquid rocket engine test

data. The steady state results when compared with nominal values show a good

agreement as a proof of the accuracy of the library.

The injector head model has been tested with a realistic test case. The new

structure of the injector dome allows to take into account the strong interaction

between the combustion chamber, the propellants in the injector dome and the

injector dome walls, evaluating the transient heat Wuxes which rise during the

ignition of an engine. The implementation of this new model has a fundamental

importance for the correct representation of the two-phase Wow inside the injector

dome and the mass Wow evolution during start-up and shut-down.

Hot-gas-side heat Wuxes in combustion devices are now described in a more

detailed way, making use of diUerent correlations for the evaluation of the hot-

gas-side heat transfer coeXcient hc. The presence of diUerent correlations and the

possibility to choose diUerent and Vne tuned correction factors allows the study

of a propulsion subsystem varying the heat Wux behaviour calculation.

Representation of thermal stratiVcation inside high aspect ratio cooling chan-

nels, and its development along transient conditions required a modiVcation of the

basic one-dimensional equations for pipes, combining semi-empirical correlations

and a Quasi-2D approach in order to save computational time and therefore keep

the model useful for system simulation purposes.

The quality and the robustness of this model has been proved Vrst comparing

its results with CFD numerical test case, and then with experimental results from

a test campaign especially performed for the evaluation of thermal stratiVcation

198

7. Conclusions

in this kind of cooling channels.

The modelling capabilities at system level have been deeply demonstrated with

the development and the creation of a model for the an entire liquid propellant

rocket engine, the RL-10A-3-3A. The construction of a model to simulate start-up

and shut-down phases of this engine required the investigation of all the main

critical aspects which occur during transient phases for all the components that

assemble the engine. Simulations for the engine pipelines, throttle and regulation

valves, turbine and pump assemblies, cooling channels and combustion chamber

have been performed to verify the correct behaviour of the components and of the

subsystems when compared with actual data.

Comparison of the transient behaviour of the engine during ground test and

model predictions is very satisfactory. Although many uncertainties aUect the

transient simulation (such as valve discharge coeXcient uncertainties, running

shaft torque, oxidiser control valve behaviour, initial conditions uncertainties etc.)

the model correctly reproduces the main phenomena occurring during transients,

such as combustion, heat transfer, turbopump operation phase change, valve

manoeuvering and pressure drops, as well as the thermodynamic behaviour of

the Wuids. Two phase Wow eUects in the engine are also well estimated. Moreover

the RL-10A-3-3A model accurately predicts the engine time-to-accelerate when

compared to ground test data.

The models developed and the simulations performed at component level and at

system level, and the understanding gathered during the analysis of the transient

phases of the RL-10 engine stimulate for further improvements and developments

to increase the reliability of such a tool for prediction and evaluation of the

transient phases of a liquid rocket engine propulsion system.

Developments could include the following:

• Taking into account the injected liquid phase into the combustion chamber

• Implementation of a chemical kinetics algorithm for Vnite rate combustion

model

199

7. Conclusions

• Development of a Vlm cooling model into the combustion chamber

• Implementation of a fully transient model for the nozzle component

• Inclusion of heat transfer and mass capacitance eUects into turbopump

models

• Inclusion of a cavitation model for pumps

In conclusion, the deep investigation in the characteristic problems that may

occur during transient phases of a liquid rocket engine and the work performed

in this thesis have brought to the development of more accurate and complex

models to evaluate peculiar phenomena inside liquid propulsion systems and the

identiVcation of additional work in order to have in the future a very reliable tool

for the prediction of liquid rocket engines start-up and shut-down phases.

200

Appendices

a

A. Implementation of Up-wind Roe Scheme

A.1. Governing equations

Here is recalled the set of governing equations that will be used in the fluid_flow_1d

library of EcosimPro. They are derived from the 1D Navier-Stokes equations,

using the conservative set of variables u = (ρ, ρu, ρE) :

∂ρ

∂t+ κwρ

∂p

∂t+∂ρu

∂x= 0

∂ρu

∂t+∂(ρu2 + p)

∂x= −Fw − ρg (A.1)

∂ρE

∂t+∂ρuH

∂x= Qw

where :

• The geometry can be quasi-1D : cross-section A can smoothly vary, so

the set of variables should change to uA = (ρA, ρuA, ρEA). Using these

variables an extra term accounting for the cross-section variation arises

in the momentum equation. This particular case of variable cross-section

is studied in the section (A.3.3). The general case is considered to be the

constant cross-section formulation (A.1);

• In order to simulate accurately the water hammer eUect, the wall compress-

ibility κw must be taken into account, through an extra term1 in the mass

conservation equation. The derivation of this term can be found in the

fluid_flow_1dManual [43];

• The gradient of shear stresses is represented in the momentum equation as

a source term Fw including all possible pressure losses in the component;

1implemented as a source term in a Vrst approximation.

b


• The work of shear stresses and external forces is neglected;

• There is one heat source representing the heat transfer with the wall Qw.

The system (A.1) of mass, momentum and energy conservation equations can be

applied to either :

• a one component, one phase Wuid : some gas or liquid;

• a one component, two-phase Wuid : the Wuid can undergo some phase

change. In this case all the variables, as well as thermodynamic and transport

coeXcients involved in the system (A.1), correspond to either the gas, the

liquid, or the 2-phase mixture, depending on the operating conditions. More

details are found in the fluid_flow_1dManual of EcosimPro.

A.1.1. 4-equation subset

The system of 3 equations above must also be extended to the case of a mixture of

two components, for which case the Vrst one can be either one phase or two-phase,

and the second one is always a non-condensable gas. Here an asymmetric formu-

lation is chosen, thus using the conservative set uasym = (ρ, ρnc, ρu, ρuE) rather

than usym = (ρ1, ρ2, ρu, ρuE). The resulting two-component set of governing

equations is :

∂ρ

∂t+ κwρ

∂P

∂t+∂ρu

∂x= 0

∂ρnc

∂t+∂ρncu

∂x= 0

∂ρu

∂t+∂(ρu2 + p)

∂x= −Fw − ρg (A.2)

∂ρE

∂t+∂ρuH

∂x= Qw

Here these equations govern the conservation of the mixture mass, non-condensable

mass, mixture momentum and mixture energy, respectively. Details about the mix-

ing rules can be found in the fluid_flow_1dManual [43] and are also recalled

in the next chapters.

c


A.2. Numerical concepts

Roe’s Wux diUerence splitting (FDS) method with the MUSCL-TVD scheme is

the most famous numerical scheme applied to enhance the numerical stability,

especially for steep gradients in density and pressure near the gas-liquid interface.

We intend to use that compressible scheme for all models under consideration,

because even the liquid Wows are inWuenced here by compressibility eUects. Some

signiVcant test cases will be performed in order to verify the accuracy of this

method.

In this section the Roe scheme is described, then the MUSCL reconstruction

methodology is given. Afterwards, a few words on the variable cross-section

formulation and consequences are given.

A.2.1. Roe’s numerical scheme

Considering hereafter the set of n general equations in matrix conservative form,

describing the behaviour of a Wuid:

∂u∂t

+∂f(u)

∂x= S(u) (A.3)

Equivalently, in quasi-linear form

∂u∂t

+ J(u) · ∂u∂x

= S(u) (A.4)

where u is a set of n conservative variables, f(u) is the conservative Wux, S(u) is

the source vector containing all terms that cannot be expressed in conservative

form, and J(u) is the Jacobian of the system deVned by :

J(u) ≡ ∂f(u)

∂u(A.5)

We focus our interest here to the convective part of (A.3), namely

∂u∂t

+∂f(u)

∂x= 0 (A.6)

d


or equivalently∂u∂t

+ J(u) · ∂u∂x

= 0 (A.7)

which form a well-deVned initial-valued hyperbolic problem provided that the

Jacobian matrix J(u) has real eigenvalues and that some initial value u(x, 0) =

u0(x) is given.

A.2.2. Approximate Riemann Solver

Several upwind diUerencing schemes, based on a cell-centered (collocated) 1D

Vnite volume formulation, have been developed to solve the conservation equa-

tions (A.6) in each control cell i = [i− 12 , i + 1

2 ] in integral form. Therefore, the

spatial evolution of the conservative variables u is piece-wise constant (constant

on each cell i), and the Wux expressions have to be evaluated at each cell interface

i+ 12 = [i, i+ 1] = [L,R] (and similarly at i− 1

2 ), as the discretisation of the Wux

is :∂f∂x

∣∣∣∣i

≈fi+ 1

2− fi− 1

2

∆x(A.8)

The interface separates the ‘left’ state denoted by uL and the ‘right’ state denoted

by uR. This deVnes precisely a non-linear Riemann problem. Starting from Go-

dunov’s original scheme [54], those schemes attempt thus to build the solution of

(A.6) by solving a succession of Riemann problems on each cell interface of the

1D domain.

Recall that the Riemann problem is the initial-value problem for (A.7) with a

discontinuous initial condition across the interface :

u(x, 0) ≡

uL x < 0

uR x > 0(A.9)

Numerical eXciency justiVes the use of a linearisation of that Riemann problem.

We concentrate here on the Approximate Riemann Solver introduced by Roe [123],

which exploits the fact that we can easily solve the Riemann problem for any

linear system of equations. So rather than solving the exact Riemann problem at

the interface, which is CPU-time consuming, we solve exactly the approximate

e


Riemann problem derived by replacing (A.7) by the local linearisation

∂u∂t

+ J(uR,uL) · ∂u∂x

= 0 (A.10)

In that case, the interface Wux fi+ 12(or fi− 1

2) can be written as :

f(uL,uR) = f(uL) + J−(uL,uR)(uR − uL) (A.11)

= f(uR)− J+(uL,uR)(uR − uL) (A.12)

=1

2

(f(uL) + f(uR)

)− 1

2|J(uL,uR)|

(uR − uL

)(A.13)

where J±(uL,uR) are the positive and negative parts of the so-called Roe-matrix

J(uL,uR), which must be constructed to satisfy the following set of conditions

christened by Roe as ‘Property U’:

i) J(uR,uL) has real eigenvalues and a corresponding complete set of linearly

independent eigenvectors;

ii) J(uR,uL)→ J(u) as uL,uR → u;

iii) J(uR,uL) must satisfy the relation :

∆f = J(uR,uL)∆u (A.14)

where the operator ∆(·) = (·)R − (·)L represents the jump in the quantity

(·) across the interface between left and right states.

Condition (i) ensures that the problem (A.10) is hyperbolic and solvable. Condi-

tion (ii) guarantees that the scheme gives satisfactory results for smooth Wows.

Condition (iii) ensures that the scheme is conservative and that the approximate

solution is coincident with the exact one when the left and right states are con-

nected by a single jump satisfying the Rankine-Hugoniot conditions (accurate

shock resolution).

Initially, Roe derived the matrix J(uR,uL) for a perfect gas as

J(uR,uL) = J(q) (A.15)

f


that is, the exact Jacobian matrix but evaluated at the so-called Roe-average state

q, which itself is an arithmetic average between left and right states, but deVned

on a parameter vector w :

Roe(q) ≡ q = q(wL + wR

2

)(A.16)

The Roe parameter w is deVned such that u and f(u) are both quadratic functions

of w. The chosen notation intends to emphasize that the average state implies

only those variables that explicitly appear in the Jacobian matrix. It is easy to

check, in this case, that (A.15), obtained by satisfying property (iii), meets all of

the other requirements set by Property U.

Roe’s original result was dedicated to the Euler equations with perfect gases,

but it has been used by several authors to achieve a simpler way of determining

J(uR,uL) for more complex systems and with other Equations of State (EoS). If

one assumes that (A.15) holds, it is possible to look immediately for the average

state q that satisVes property (iii) by direct substitution in (A.14) or in the eigen-

vector expansion of ∆f and ∆u. Surprisingly, an exact deVnition of a Roe-average

for non-perfect gases not only exists but is actually not unique. All the methods

cited above lead to a matrix J(uL,uR) involving undeVned coeXcients, which are

the Roe-average pressure derivatives. More details can be found in the following

chapters.

Eigen decomposition

We recall that in the numerical Wux expression (A.13), the absolute value of the

Roe-matrix J(uR,uL) is needed. For a given matrix, say the Jacobian J , the

absolute value of J is deVned through its diagonalization as:

|J | = R · |Λ| · L (A.17)

where L and R are the left and right eigenmatrices respectively, and |Λ| containsthe absolute values of the eigenvalues λk of J on its diagonal. These eigenvalues

g


can be found by solving :

|J − λI| = 0 (A.18)

for λ. The absolute eigenvalue diagonal matrix is then :

|Λ| =

|λ1| 0 · · · 0

0 |λ2| · · · 0... . . . ...

0 . . . 0 |λn|

(A.19)

The right eigenvectors Rk forming the columns of the n × n right eigenmatrix

R = (R1R2 · · ·Rn) are found by solving these n systems : J · Rk = λkRk. Like-

wise, the left eigenvectors Lk forming the rows of the n × n left eigenmatrix

L = (L1,L2, · · · ,Ln) are found by solving these n systems: Lk · J = λkLk. Con-

sistency imposes that L ·R = I , where I is the n× n identity matrix.

In the same way, the absolute value of the Roe-matrix J(uR,uL) is derived:

|J(uR,uL)| = R · |Λ| · L (A.20)

Following property (A.15) of Roe’s scheme, stating that the linearised Jacobian

is the exact Jacobian but evaluated at some Roe-average state q, the average

eigenvalues and eigenmatrices are the exact ones but evaluated at that Roe-

average:

L = L(q) (A.21)

R = R(q) (A.22)

Λ = Λ(q) (A.23)

Now if we project the conservative variable diUerence onto the right eigenvectors

Rk

∆u = uR − uL = R · a (A.24)

h


from which one Vnd the wave strengths

a = (α1, α2, ..., αn)t

= R−1 ·∆u

= L ·∆u (A.25)

We can now explicitly show the eigen decomposition in the diUusive term of

(A.13):

|Ji+ 12| ·∆u = R · |Λ| · L ·∆u

= R · |Λ| · a

=n∑

k=1

αk|λk|Rk

(A.26)

Recall that the numerical interface Wux f(uL,uR) is given by (A.13). Using the

above eigen decomposition, we have Vnally

f(uL,uR) =1

2

(f(uL) + f(uR)

)− 1

2

n∑k=1

αk|λk|Rk

(A.27)

This shows that this Roe-matrix J(uR,uL) is used as a characteristic-based con-

trolled amount of numerical diUusion.

A.3. Reconstruction method

A.3.1. Higher order accuracy

The order of accuracy of the scheme presented is however Vrst order : the variables

ui are still constant within each cell. We can retrieve a higher order accuracy

scheme by reconstructing the variations ui(x) on each cell, through the MUSCL

approach (Monotone Upstream-centred Scheme for Conservation Laws).

A possibly piecewise quadratic local reconstruction of ui(x) within cell [i− 12 , i+

12 ]

is:

ui(x) = ui +x− xi

∆xδ(1)ui +

3ω

2∆x2

[(x− xi)

2 − ∆x2

12

]δ(2)ui (A.28)

i


where ω ∈ [−1, 1] is a free parameter (see Table A.1) and δ(1/2)ui an estimation

of the Vrst/second derivative of ui(x), respectively. Remark that the nodal value

ui(x = xi) is not necessarily equal to ui:

ui(x = xi) = ui −ω

8δ(2)ui (A.29)

If we require these gradients δ(1/2)ui to depend only on adjacent cells, we have

simply:

δ(1)ui = 12(ui+1 − ui−1) =

1

2(∆u1st

i+ 12

+ ∆u1sti− 1

2

) (A.30)

δ(2)ui = ui+1 − 2ui + ui−1 = ∆u1sti+ 1

2

−∆u1sti− 1

2

(A.31)

where the following notations for the jumps between constant values have been

introduced:

∆u1sti− 1

2

≡ ui − ui−1 (A.32)

∆u1sti+ 1

2

≡ ui+1 − ui (A.33)

Actually, the resolution of the approximated Riemann problem requires only the

values at the cell boundaries i± 12 , extrapolated from (A.28):

u+i− 1

2

≡ ui(x = xi− 12) = ui −

1

2δ(1)ui +

ω

4δ(2)ui (A.34)

u−i+ 1

2

≡ ui(x = xi+ 12) = ui +

1

2δ(1)ui +

ω

4δ(2)ui (A.35)

where the ± superscripts denote the right/left side of the interface, respectively.

With the slope deVnitions (A.30)-(A.31), the extrapolated boundary values at the

interfaces i± 12 become:

u−i− 1

2

= ui−1 +1

4(1− ω)∆u1st

i− 32

+1

4(1 + ω)∆u1st

i− 12

(A.36)

u+i− 1

2

= ui −1

4(1 + ω)∆u1st

i− 12

− 1

4(1− ω)∆u1st

i+ 12

(A.37)

u−i+ 1

2

= ui +1

4(1− ω)∆u1st

i− 12

+1

4(1 + ω)∆u1st

i+ 12

(A.38)

u+i+ 1

2

= ui+1 −1

4(1 + ω)∆u1st

i+ 12

− 1

4(1− ω)∆u1st

i+ 32

(A.39)

j


The high-order reconstructed jump at the interface i+ 12 is given by:

∆uhoti+ 1

2

≡ u+i+ 1

2

− u−i+ 1

2

=1

4(1− ω)(−ui+2 + 3ui+1 − 3ui + ui−1) (A.40)

= −1

4(1− ω)

∂3ui

∂x3+O(∆x4) (A.41)

Depending on the value of ω, diUerent schemes and orders of accuracy can be

reached. The following table summarizes the diUerent choices.

ω Reconstruction Order Nodal value, see (A.29) Jump value, see (A.40)-1 linear one-sided 2nd 1

8(ui−1 + 6ui + ui+1) 12(−ui+2 + 3ui+1 − 3ui + ui−1)

0 linear up/down 2nd ui14(−ui+2 + 3ui+1 − 3ui + ui−1)

13 parabolic 3rd 1

24(−ui−1 + 26ui − ui+1) 16(−ui+2 + 3ui+1 − 3ui + ui−1)

1 linear central 2nd 18(−ui−1 + 10ui − ui+1) 0

Table A.1.: DiUerent values of ω.

Some remarks on this table:

• We can see that for ω = −1, the interpolation is fully one-sided, as the

extrapolated boundary values are computed using two upstream cells;

• Using ω = 0, that extrapolation uses one upstream and one downstream cell.

Moreover, only for ω = 0 do we have a nodal value equal to the constant

value ui;

• Only for ω = 13 do we have a parabolic interpolation, and thus a third order

accuracy scheme. Indeed for ω = 13 , the reconstruction (A.28) is a correct

Taylor development up to the third order;

• Using ω = 1, the scheme looses its upwind behaviour, as the interpolation is

a simple arithmetic average between adjacent cells. The scheme corresponds

to a central scheme as there is no discontinuity at the interface : the jump

value is zero.

k


An example of piecewise linear reconstruction with ω = 0 is shown on Fig.A.1. We

see that without reconstruction, the jump at the interface i+ 12 is ∆u1st

i , whereas

after a linear reconstruction, the jump at the interface is ∆u2ndi .

Figure A.1.: Piece-wise linear reconstruction.

l


High-resolution scheme

A typical problem with a higher-order accurate discretisation is the spurious

oscillations, which appear in the vicinity of the non-smooth solutions (Godunov’s

theorem). The problem is solved if a combination of the Vrst- and the higher-order

accurate discretisation is used.

Therefore, the higher order data reconstruction (A.28) is constrained through

a TVD version (Total Variation Diminishing) of this approach to retrieve a Vrst-

order scheme near strong gradients, and able a higher-order scheme in smooths

parts of the Wow. The slopes δ(1/2)ui must be limited, and therefore the extrapo-

lated boundary values have the following limited expression:

u−i− 1

2

= ui−1 +1

4(1− ω)φ+

i− 32

∆u1sti− 3

2

+1

4(1 + ω)φ−

i− 12

∆u1sti− 1

2

(A.42)

u+i− 1

2

= ui −1

4(1 + ω)φ+

i− 12

∆u1sti− 1

2

− 1

4(1− ω)φ−

i+ 12

∆u1sti+ 1

2

(A.43)

u−i+ 1

2

= ui +1

4(1− ω)φ+

i− 12

∆u1sti− 1

2

+1

4(1 + ω)φ−

i+ 12

∆u1sti+ 1

2

(A.44)

u+i+ 1

2

= ui+1 −1

4(1 + ω)φ+

i+ 12

∆u1sti+ 1

2

− 1

4(1− ω)φ−

i+ 32

∆u1sti+ 3

2

(A.45)

In these expressions, the limiting coeXcients φ±i± 1

2

and φ∓i± 3

2

are deVned as:

φ−i− 1

2

= φ(r−i− 1

2

) φ+i− 1

2

= φ(r+i− 1

2

) (A.46)

φ−i+ 1

2

= φ(r−i+ 1

2

) φ+i+ 1

2

= φ(r+i+ 1

2

) (A.47)

φ−i+ 3

2

= φ(r−i+ 3

2

) φ+i− 3

2

= φ(r+i− 3

2

) (A.48)

and the r function ‘measures’ the smoothness of the solution, as it is deVned as a

ratio of consecutive variations:

r−i− 1

2

=∆u1st

i− 32

∆u1st

i− 12

=1

r+i− 3

2

(A.49)

r+i− 1

2

=∆u1st

i+12

∆u1st

i− 12

=1

r−i+ 1

2

(A.50)

r+i+ 1

2

=∆u1st

i+32

∆u1st

i+12

=1

r−i+ 3

2

(A.51)

m


The function φ can be any slope limiter, see [85] for details. It has been decided

that both MinMod φmm and SuperBee φsb limiters will be used and tested. They

are deVned as:

φmm(r) = max [0,min(1, r)] (A.52)

φsb(r) = max [0,min(2r, 1),min r, 2] (A.53)

In that way, the scheme is second- or possibly third-order accuracy in space in

the smooth parts of the Wow, and reduces to Vrst order where strong gradients

appear. This so-called high-resolution scheme should now be oscillation-free near

discontinuities.

In summary, instead of using uR = ui+1 and uL = ui at the cell interface

i+ 12 , it is used uR = u+

i+ 12

and uL = u−i+ 1

2

, as deVned in (A.44)-(A.45), so that the

high-resolution reconstructed jump at the interface i+ 12 is given by:

∆uhri+ 1

2

≡ u+i+ 1

2

− u−i+ 1

2

(A.54)

In other words, the explicit numerical Wux was Vrst order :

f 1sti+ 1

2=

1

2

(f(ui+1) + f(ui)

)−|Ji+ 1

2|

2·∆u1st

i+ 12

(A.55)

And it is now :

f hri+ 1

2=

1

2

(f(u+

i+ 12

) + f(u−i+ 1

2

))−|Ji+ 1

2|

2·∆uhr

i+ 12

(A.56)

n


A.3.2. Preconditioning

Preconditioning is a procedure used to deal with the typical stiUness prob-

lems [144, 139] encountered when a compressible solver like Roe’s scheme is

used to solve the governing equations of a Wow in the low Mach number range.

DeVning as usual the Mach number M as the ratio of Wow velocity u to sound

speed c, the low Mach number region is roughly deVned asM ≤ 0.2.

In two-phase Wows, liquid phases are likely to be in the low Mach number region.

Indeed in liquids the sound speed is generally by far greater than the Wow velocity.

Similarly, the vapor phase is generally outside this low Mach number range, unless

there is possible reverse Wow phenomena. Moreover, strong variations of the Mach

number are likely to occur at phase transitions, or when the cross-section of the

component is not constant.

Therefore the compressible Roe scheme, suited for M ≥ 0.2 approximatively,

must be modiVed accordingly as regions with M ≤ 0.2 and M ≥ 0.2 occur si-

multaneously. Since the magnitude of the sound speed relative to that of the

Wow velocity is responsible for the stiUness of the compressible Wow equations

at low Mach numbers, this problem is dealt with by artiVcially scaling down

the amplitude of the acoustic waves in order to improve the conditioning of the

system : this is know as preconditioning.

This can be achieved by some algebraic manipulation of the time derivative

terms and the Roe-matrix of the original system of equations. A better condi-

tioning of the system leads to improved accuracy and convergence in steady and

unsteady computations.

The stiUness problem is solved by multiplying the time-derivative of the sys-

tem of equations (A.3) by the preconditioning matrix P−1 :

P−1∂u∂t

+∂f∂x

= 0 (A.57)

o


In that way, the convergence to steady-sate is accelerated, but the time consistency

is lost. An alternative formulation, performing a correct scaling of the artiVcial

dissipation terms, but keeping the original form (A.3) of the equations, enables a

time-accurate unsteady computation: rather than using (A.13) for the numerical

Wux, it is used:

fpre(uL,uR) =1

2

(f(uL) + f(uR)

)− 1

2P−1 · |P · J(uL,uR)| ·∆u (A.58)

The form of that preconditioning matrix P is dependent on the model considered,

and therefore the reader is referred to the corresponding sections of each model,

where its explicit formulation is given. A deeper analysis of preconditioning

methods can be found, for instance, in [139, 81].

A.3.3. Variable cross-section

When the cross-section of the component varies radially along the longitudinal

axis, the conservation laws (A.1) or (A.2) must be modiVed. A new set of conser-

vative variables is used : uA = A(ρ, ρu, ρE), and the governing set of equations is

:

∂ρA

∂t+ Aκwρ

∂p

∂t+∂ρuA

∂x= 0(∂ρncA

∂t+∂ρncuA

∂x= 0

)if needed

∂ρuA

∂t+∂(ρu2 + p)A

∂x= p

dA

dx− AFw − Aρg (A.59)

∂ρEA

∂t+∂ρuHA

∂x= AQw

In order to keep these equations under conservative form, an extra source term

appears in the momentum equation, explicitly showing the inWuence of the cross-

section variation.

Apart from this extra source term, only a matter of notations distinguishes this set

of equations from the general one (constant cross-section) : the equations remain

unchanged. Consequently, only the constant cross-section case is described in

the next chapters, unless speciVed otherwise. One has only to remind that an

p


extra term should be added if the cross-section were to vary along the axis of the

component.

A way to deal with variable cross-section in EcosimPro could be the following :

• Implement the above formalism to the component continuous block (most

likely the pipe);

• Build another component inherited from the Vrst one, where the change of

cross-section is deVned : either the change is zero (constant cross-section) or

the area varies with the longitudinal direction following a given expression

(the smoother the better).

q

B. Friction Factor Correlations

B.1. Single-Phase Friction Factor Calculation. Function

hdc_fric

The function hdc_fric incorporates the evolution of the friction factor as a

function of the local Reynolds number (ρvD/µ) and the roughness ε.

The friction factor (f ) is calculated by means of a simple correlation valid for

laminar, turbulent and transient Wow.

f = 8 ·

[(8

Re

)12

+1

(A+B)3/2

] 112

(B.1)

where

A =

[2.457 ln

1

(7/Re)0.9 + 0.27ε/D

]16

B =

[37530

Re

]16

B.2. Two-Phase Friction Factor Calculation. Friedel

Correlation

The following formulation is taken from reference [135].

The correlation method of Friedel [48] (1979) utilizes a two-phase multiplier:

∆Pfrict = ∆Pl Φ2fr (B.2)

r


where ∆Pl is calculated for the liquid-phase Wow as:

∆Pl = 4fl

(L

D

)G2

(1

2ρl

)(B.3)

The liquid friction factor fl and liquid Reynolds number are obtained from

f =0.079

Re0.25Re =

GD

µ(B.4)

Using the liquid dynamic viscosity µl. His two-phase multiplier is

Φ2fr = E +

3.24F H

Fr0.045H We0.035

l

(B.5)

The dimensionless factors FrH , E, F and H are as follows:

FrH =G2

g D ρ2H

E = (1− x)2 + x2 ρl fg

ρG fl

F = x0.78(1− x)0.224

H =

(ρl

ρg

)0.91(µG

µL

)0.19(1−

µg

µl

)0.7

The liquid WeberWel is deVned as

Wel =G2D

σ ρH(B.6)

where σ is the surface tension. The following alternative deVnition of the homo-

geneous density ρH based on vapor quality is used:

ρH =

(x

ρG+

1− xρL

)−1

(B.7)

B.3. Elbow Pressure Loss Function

This function calculates the bend pressure drop coeXcient. It depends on the

relative radius of curvature, Rbend/D, the relative roughness, ε/D, and the bend

s


angle, α.

According to Idelchik [63], the total resistance coeXcient of pipe bends is the

product of the following coeXcients (see Figure B.1):

Figure B.1.: Elbow pressure loss parameters

• angle eUect:

ξangle = 0.957α

90+ 0.226

√α

90+ 0.407 sin(α)− 0.833 sin(α/2)

• radius eUect:

ξradius =

0.21/

√R

D(R/D > 1)

0.21/

(R

D

)2.5

(R/D < 1)

• roughness eUect:

ξroug =

min(2, 1 + 106

( εD

)2) (R/D > 1.5)

min(2, 1 + 103( εD

)) (R/D < 1.5)

t


Then the pressure drop coeXcient is:

ξbend = ξangle · ξradius · ξroug (B.8)

u

C. Film CoeXcient Calculation

The Vlm coeXcient h is evaluated by mean of the Nusselt number calculation.

The correlation used for the Nusselt assessment is function of the quality of the

Wuid, therefore there will be a speciVc correlation for single-phase Wuid, two-phase

Wuid etc..

Single phase :

Laminar and turbulent Nusselt numbers:

Nulam = 4 (C.1)

Nutur = 0.023Re0.8 Pr0.4 (C.2)

The equivalent Nusselt number covering transitions zones is:

Nu = (Nu16lam +Nu16

tur)1/16 (C.3)

Then, the single phase Vlm coeXcient is calculated as follows:

hsp = Nu

(λ

D

)(C.4)

Condensation (two-phase or vapour and Tw < Tsat) :

This method is based on Boyko & Kruzhilin’s [37] correlation, appropriate for

Vlm-wise condensation in uniform channels under forced convection conditions:

hcond = hsp

√1 + x(ρl/ρg − 1) (C.5)

Superheated Condensation (Quality =1 and Tw < Tsat) :

The method used for the calculation of the heat transfer coeXcient is shown below.

v


The method was chosen in order to keep a continuity between single and two

phase regimes:

h =hg (Tsat − Tw) + hcond (T − Tsat)

(T − Tw)(C.6)

where hg is the vapour single phase Vlm coeXcient. hcond is the condensation

correlation Vlm coeXcient considering the actual pressure saturation properties.

Boiling (Quality < 0.7 and Tw > Tsat) :

According to Chen [26], for vapour quality < 0.7 and if the stratiVcation is not

severe, several steps must be followed in order to calculate the Vlm coeXcient

under vaporization regime. First of all, the convective Vlm coeXcient for liquid

must be calculated:

hl = 0.023Re0.8l Pr0.4

l

(λl

D

)(C.7)

where the sub index l makes reference to saturated liquid conditions. The inverse

of the Lockhart-Martinelli [89] parameter is calculated:

1/Xtt =

(x

(1− x)

)0.9(ρl

ρg

)0.5(µg

µl

)0.1

(C.8)

Then, the convective boiling contribution is calculated as follows: hcon = F hl ,

where:

F =

2.35 (1/Xtt + 0.213)(0.736) (1/Xtt > 0.1)

1 (1/Xtt > 0.1)

The nucleate boiling contribution is calculated as follows:

hnuc = B (Tw − Tsat)0.24 (Psat,Tw − Psat)

0.75 (C.9)

where

B = 0.001222λ0.79

l Cp0.45l ρ0.49

l S

σ0.5 µ0.29l (ρg(hg − hl))0.24

(C.10)

S =1

1 + 2.53 exp−6Re1.172ph

Re2ph = Rel, F1.25 (C.11)

w


where σ is the surface tension; hg − hl is the vaporization latent heat. Finally, the

combined boiling Vlm coeXcient is hChen = hcon + hnuc

Boiling (Quality > 0.9 and Tw > Tsat) :

For vapour quality x > 0.9 a post-dry-out correlation due to Dougall & Rohsenow [37]

is used:

hg = 0.023Re0.8g Pr0.4

g

(λg

D

)(C.12)

Φ = x+ (1− x)ρg

ρl(C.13)

hDR = hg · Φ (C.14)

Boiling (0.7 < Quality < 0.9 and Tw > Tsat) :

For vapour quality 0.7 < x < 0.9, cubic spline interpolation is performed between

the Chen & Dougall - Rohsenow correlations.

Subcooled Boiling (Quality = 0 and Tw > Tsat) :

The method used is to calculate the heat transfer coeXcient as to ensure continuity

between the single and two phase regimes:

h =hl (Tsat − Tl) + hChen (Tw − Tsat)

(Tw − T )(C.15)

where hl is the liquid single phase Vlm coeXcient. hChen is the Chen correlation

Vlm coeXcient considering the actual pressure saturation properties.

x

Bibliography

[1] R. Abgrall. How to Prevent Pressure Oscillations in Multicomponent Flow

Calculations: a Quasi Conservative Approach. J. Comput. Phys., 14:150–160,

1996.

[2] A. Adamkowski and M. Lewandowski. Experimental Examination of Un-

steady Friction Models for Transient Pipe Flow Simulation. Journal of

Fluids Engineering, 128:1351 – 1363, November 2006.

[3] G. Albano, J. Hebrard, and V. Leudiere. HM7B Engine Transient Simulator

with CARINS tool. In EUCASS, 2005.

[4] M. A. Arguello. The Concept Design of a Split Flow Liquid Hydrogen

Turbopump. Master’s thesis, Air Force Institute of Technology, March 2008.

[5] Astrium and TU Dresden. Final report on data correlation and evaluation

of test results phase 1 part 1. Internal ESA contractor report GSTP-2-TN-06-

Astrium, Astrium, 2001.

[6] M. Atsumi, A. Ogawara, K. Akazawa, and K. Takeishi. Development of

Visual Integrated Simulator for Rocket Engine Cycle. In 4th International

Conference on Launcher Technology, Space Launcher Liquid Propulsion,

Belgium, 2002. CNES.

[7] T. J. Avampato and C. Saltiel. Dynamic Modeling of Starting Capabilities

of Liquid Propellant Rocket Engines. Journal of Propulsion and Power,

Vol.11(No.2):pp.292 – 300, March - April 1995.

[8] F. Azevedo. Propriétés thermodynamiques des ergols. PhD thesis, CNES.

i

Bibliography

[9] D. R. Bartz. Turbulent boundary-layer heat transfer from rapidly acceler-

ating Wow of rocket combustion gases and of heated air. Technical Report

NASA-CR-62615, Jet Propulsion Laboratory, 1963.

[10] E.N. Belyaev, V.K. Chvanov, and V.V. Chervak. Mathematical Modelling

of the WorkWow of Liquid Rocket Engines. MAI, 1999. (In Russian).

[11] R. Biggs. History of Liquid Rocket Engine Development in the United

States, 1995 - 1980, volume Vol. 13. American Astronautical Society, 2001.

[12] R. Biggs. Space Shuttle Main Engine: the First Twenty Years and Beyond,

volume Vol. 29. American Astornautical Society, 2008.

[13] M. Binder. An RL10A-3-3A Rocket Engine Model Using the Rocket Engine

Transient Simulator (ROCETS) Software. Contractor Report NASA CR-

190786, NASA, July 1993.

[14] M. Binder. A Transient Model of the RL10A-3-3A Rocket Engine. Contractor

Report NASA CR-195478, NASA, July 1995.

[15] M. Binder, T. Tomsik, and J.P. Veres. RL10A-3-3A Rocket Engine Modeling

Project. Technical Memorandum NASA TM-107318, NASA, 1997.

[16] M. P. Binder and J. L. Felder. Predicted Performance of an Integrated

Modular Engine System. InAIAA/SAE/ASME/ASEE 29th Joint Propulsion

Conference and Exhibit, number AIAA 1993-1888, 1993.

[17] J. E. Bradford, A. Charania, and B. St. Germain. REDTOP-2: Rocket Engine

Design Tool Featuring Engine Performance, Weight, Cost, and Reliability. In

40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit,

number AIAA 2004-3514, 2004.

[18] K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-

Value Problems in DiUerential-Algebraic Equations. SIAM, 1996.

[19] C. D. Brown. Conceptual Investigations for a Methane-Fueled Expander

Rocket Engine. In 40th AIAA/ASME/SAE/ASEE Joint Propulsion Confer-

ence and Exhibit, number AIAA 2004-4210, 2004.

ii

Bibliography

[20] J. R. Brown. Cryogenic Upper Stage Propulsion - RL10 and Derivative

Engines. Technical report, Pratt&Whitney, 1990.

[21] J. R. Brown, R. R. Foust, D. E. Galler, P. G. Kanic, T. D. Kmiec, C. D. Limerick,

R. J. Peckham, and T. Swartwout. Design and Analysis Report for the RL10-

2B Breadboard Low Thrust Engine. Contract Report NASA-CR-174857,

NASA, 1984.

[22] B. Campbell and R. Davis. Quasi-2D Unsteady Flow Solver Mod-

ule for Rocket Engine and Propulsion System Simulations. In 42nd

AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, num-

ber AIAA 2006-5061, 2006.

[23] Van P. Carey. Liquid-Vapor Phase-Change Phenomena. Taylor and Fran-

cis, 1992.

[24] J. Carlile and R. Quentmeyer. An Experimental Investigation of High-

Aspect-Ratio Cooling Passages. In AIAA/SAE/ASME/ASEE 28th Joint

Propulsion Conference and Exhibit, number AIAA 1992-3154, July 1992.

[25] H. Chaudhry. Applied Hydraulic Transients - 2nd Edition. Van Nostrand

Reinhold Company, New York, 1987.

[26] J. C. Chen. A correlation for boiling heat transfer to saturated Wuid in

convective Wow. I.Eng. Chem. Process Des. Dev., 5:322–329, 1966.

[27] S. Clerc. Numerical Simulation of the Homogeneous Equilibrium Model

for Two-Phase Flows. Journal of Computational Physics, Vol. 161:pp. 354

– 375, 2000.

[28] Kenneth P. CoXn and Cleveland O’Neal, Jr. Experimental thermal con-

ductivities of the N2O4 - 2NO2 system. Technical Note TN-4209, NASA,

1958.

[29] A. M. Crocker and S. Peery. System Sensitivity Studies of a LOX/Methane

Expander Cycle Rocket Engine. In 34th AIAA/ASME/SAE/ASEE Joint

Propulsion Conference and Exhibit, number AIAA 1998-3674, 1998.

iii

Bibliography

[30] A. Dalbies, G. De Spiegeleer, C. Maquet, and E. Robert. Engines Trade OU for

Ariane 2010 Initiative. In AIAA/ASME/SAE/ASEE 38th Joint Propulsion

Conference and Exhibit, number AIAA 2002-3839, 2002.

[31] R. Davis and B. T. Campbell. Quasi-One-Dimensional Unsteady-Flow

Procedure for Real Fluids. AIAA Journal, Vol.45(10):pp.2422 – 2428, October

2007.

[32] M. De Rosa, J. Steelant, and J. Moral. ESPSS: European Space Propulsion

System Simulation. In Space Propulsion Conference, May 2008.

[33] J.C. DeLise and M.H.N. Naraghi. Comparative Studies of Convective Heat

Transfer Models for Rocket Engines. In AIAA/ASME/SAE/ASEE 31st Joint


[34] F. Di Matteo and M. De Rosa. Object Oriented Steady State Analysis and

Design of Liquid Rocket Engine Cycles. In 3AF, editor, Space Propulsion

Conference 2010, San Sebastian, Spain, May 2010.

[35] F. Di Matteo, M. De Rosa, and M. Onofri. Semi-Empirical Heat Transfer

Correlations in Combustion Chambers for Transient System Modelling. In

3AF, editor, Space Propulsion Conference 2010, San Sebastian, Spain, May

2010.

[36] F. Di Matteo, M. De Rosa, M. Pizzarelli, and M. Onofri. Modelling of

StratiVcation in Cooling Channels and its Implementation in a Transient

System Analysis Tool. In AIAA, editor, AIAA/SAE/ASME/ASEE 46th Joint

Propulsion Conference and Exhibit, Nashville, TN, USA, July 2010. AIAA.

[37] R.S. Dougall and W.H. Rohsenow. Film boiling on the inside of the inside of

the vertical tubes with upward Wow of the Wuid at low qualities. Technical

Report 9079.26, M.I.T., 1963.

[38] S. Durteste. A Transient Model of the VINCI Cryogenic Upper Stage Rocket

Engine. In 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference

and Exhibit, number AIAA 2007-5531, 2007.

iv

Bibliography

[39] Y. Elkouch. Physical Model SpeciVcation Cenaero’s Contribution. Technical

Note TN-3221, CENAERO, 2007.

[40] Empresarios Agrupados. Ecosimpro: Continuous and discrete modelling

simulation software. http://www.ecosimpro.com, 2007.

[41] Empresarios Agrupados. ESPSS-2 industrial evaluation. Internal ESA

document, May 2009.

[42] Empresarios Agrupados. ESPSS user manual. 1.4.1 edition, 2009.

[43] Empresarios Agrupados. ESPSS user manual. 2.0 edition, 2010.

[44] Capcom Espace. Le moteur HM7B. http://www.capcomespace.net

/dossiers/espace_europeen/ariane/ariane4/moteur_HM7B.htm. retrieved

28/04/2010.

[45] Capcom Espace. Le moteur Vinci. http://www.capcomespace.net

/dossiers/espace_europeen/ariane/. retrieved 28/10/2011.

[46] Capcom Espace. Le moteur Vulcain 2. http://www.capcomespace.net

/dossiers/espace_europeen/ariane/. retrieved 28/10/2011.

[47] A. Espinosa Ramos. CARAMEL: The CNES computation sotware for design-

ing Liquid ROcket Engines. In 4th International Conference on Launcher

Technology, Space Launcher Liquid Propulsion, 2002.

[48] L. Friedel. Improved Friction Pressure Drop Correlations for Horizontal

and Vertical Two Phase Pipe Flow. In European Two Phase Flow Group

Meeting, Italy, 1979. Ispra.

[49] A. Frohlich, M. Popp, G. Schmidt, and D. Thelemann. Heat Transfer Char-

acteristics of H2/O2 - Combustion Chambers. In AIAA/SAE/ASME/ASEE

29th Joint Propulsion Conference, number AIAA 93-1826, June 1993.

[50] Y. Fukushima, H. Naktsuzi, R. Nagao, K. Kishimoto, K. Hasegawa, T. Ko-

ganezawa, and S. Warashina. Development Status of LE-7A and LE-5B

Engines for H-IIA Family. Acta Astronautica, Vol. 50:pp. 275 – 284, 2002.

v

Bibliography

[51] F. Gao, Y. Chen, and Z. Zhang. Numerical Simulation of Steady and

Filling Process of Low Temperature Liquid Propellants Pipeline. In Fifth

International Conference on Fluid Mechanics. Tsinghua University Press

& Springer, August 2007.

[52] A. J. Glassman. Computer Code for Preliminary Sizing Analysis of Axial-

Flow Turbines. Contractor Report CR-4430, NASA, 1992.

[53] S. Go. A historical survey with success and maturity estimates of launch

systems with RL10 upper stage engines. In IEEE, editor, Reliability and

Maintainability Symposium, 2008.

[54] S.K. Godunov. A DiUerence Scheme for Numerical Computation of Dis-

continuous Solutions of Equations of Fluid Dynamics. Math. Sbornik,

47(89):271–306, 1959.

[55] Sanford Gordon and Bonnie J. McBride. Computer program for calculation

of complex chemical equilibrium compositions and applications. Technical

Report RP-1311, NASA, 1994.

[56] T. M. Haarmann. Numerische Simulation des Warmeübergangs in einer

kryogenen Raketenbrennkammer. PhD thesis, RWTH Aachen, 2006.

[57] T. M. Haarmann and W. W. Koschel. Computation of Wall Heat

Fluxes in Cryogenic H2/O2 Rocket Combustion Chambers. In

AIAA/ASME/SAE/ASEE 38th Joint Propulsion Conference and Exhibit,

number AIAA 2002-3693, July 2002.

[58] M. S. Haberbusch, C. T. Nguyen, and A. F. SkaU. Modeling RL10

Thrust Increase with DensiVed LH2 and LOX Propellants. In 39th


ber AIAA 2003-4485, July 2003.

[59] M. S. Haberbusch, A. F. SkaU, and C. T. Nguyen. Modeling the RL-

10 with DensiVed Liquid Hydrogen and Oxygen Propellants. In 38th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, number

AIAA 2002-3597, July 2002.

vi

Bibliography

[60] H. Hearn. Development and Application of a Priming Surge Analysis

Methodology. In 41st AIAA/ASME/SAE/ASEE Joint Propulsion Confer-


[61] W. D. Huang. An Object-Oriented Analysis Method for Liquid Rocket

Engine System. In 36th AIAA/ASME/SAE/ASEE Joint Propulsion Confer-


[62] Dieter K. Huzel and David H. Huang. Modern Engineering for Design of

Liquid-Propellant Rocket Engines, volume 147 of Progress in Astronautics

and Aeronautics. AIAA, 1992.

[63] I. E. Idelchik. Handbook of Hydraulic Resistance. Begell House, 3rd edition,

2001.

[64] Inc.. Information Systems Laboratories. RELAP5/MOD3.3 CODE MAN-

UAL VOLUME IV: MODELS AND CORRELATIONS. Nuclear Safety

Analysis Division, Rockville, Maryland, December 2001.

[65] A. Isselhorst. HM7B Simulation with ESPSS Tool on ARIANE 5 ESC-A Up-

per Stage. In 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference


[66] K. J. Kacynski. Thermal stratiVcation potential in rocket engine coolant

channels. Technical Memorandum NASA-TM-4378, NASA, 1992.

[67] G. P. Kalmykov, E. V. Lebedinsky, V. I. Tararyshkin, and I. O. Yeliseev.

Expander LRE of 200 tf thrust on hydrocarbon fuel. In 4th International


2002.

[68] V. M. Kalnin and V. A. Sherstiannikov. Hydrodynamic Modelling of the

Starting Process in Liquid-Propellant Engines. Acta Astronautica, Vol. 8:pp.

231 – 242, 1981.

[69] V. M. Kalnin and V. A. Sherstiannikov. Vibration and Pulsation Processes in

feed Systems of Liquid Rocket Engines. Acta Astronautica, Vol. 10:pp. 713

– 718, 1983.

vii

Bibliography

[70] A. Kanmuri, T. Kanda, Y. Wakamatsu, Y. Torri, E. Kagawa, and

K. Hasegawa. Transient Anaysis of LOX/LH2 Rocket Engine (LE-7) . In

25th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, number AIAA

89-2736, July 1989.

[71] H. Karimi and R. Mohammadi. Modeling and simulation of a two com-

bustion chambers liquid propellant engine. Aircraft Engineering and

Aerospace Technology, Vol.79:pp. 390 – 397, 2007.

[72] H. Karimi, R. Mohammadi, and E. Taheri. Dynamic Simulation and Paramet-

ric Study of a Liquid Propellant Engine. In 3rd International Conference

on Recent Advances in Space Technologies, 2007.

[73] H. Karimi, A. Nassirharand, and M. Beheshti. Dynamic and Nonlinear

Simulation of Liquid-Propellant Engines. Journal of Propulsion and Power,

Vol.19(No.5):pp. 938 – 944, September - October 2003.

[74] J. Keppeler, E. Boronine, and F. Fassl. Startup Simulation of Upper Stage

Propulsion System of ARIANE 5. In 4th International Conference on

Launcher Technology, Space Launcher Liquid Propulsion, 2002.

[75] T. Kimura, M. Sato, T. Masuoka, T. Kanda, and A. Osada. EUects of Deep

Throttling on Rocket Engine Systems . In 46th AIAA/ASME/SAE/ASEE

Joint Propulsion Conference and Exhibit, number AIAA 2010-6727, 2010.

[76] A. E. Krach and A. M. Sutton. Another Look at the Practical and

Theoretical Limits of an Expander Cycle, LOX/H2 Engine. In 35th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, 1999.

[77] L. Kun and Z. Yulin. A Study on Versatile Simulation of Liquid Propellant

Rocket Engine Systems Transients. In AIAA/ASME/SAE/ASEE 36th Joint


[78] J. C. Leahy, S. G. Hanna, J. Y. Malchi, and E. S. Kim. Liquid Propulsion:

Engine Production and Operation. In Encyclopedia of Aerospace Enngi-

neering. John Wiley & Sons, 2010.

viii

Bibliography

[79] F. LeBail and M. Popp. Numerical Analysis of High Aspect Ratio Cooling

Passage Flow and Heat Transfer. In AIAA/SAE/ASME/ASEE 29th Joint

Propulsion Conference, number AIAA 1993-1829, June 1993.

[80] A. LeClair and A. K. Majumdar. Computational Model of the Chill-

down and Propellant Loading of the Space Shuttle External Tank. In



[81] D. Lee. Local Preconditioning of the Euler and Navier-Stokes Equations.

PhD thesis, University of Michigan, 1996.

[82] B. Legrand, G. Albano, and P. Vuillermoz. Start up transient modelling

of pressurised tank engine: AESTUS application. In 4th International


2002.

[83] A. J. Letson and C. Christian. RL10A-3-3 Rocket Engine Oxidizer Pump

Development Program. Contract Report NASA-CR-83547, NASA, 1966.

[84] V. Leudiére, P. Supié, and M. Villa. KVD1 Engine in LOX/CH4. In

AIAA/ASME/SAE/ASEE 43rd Joint Propulsion Conference & Exhibit,

number AIAA 2007-5446, 2007.

[85] R.J. Leveque. Finite Volume Methods for Hyperbolic Problems. Cambridge

Texts in Applied Mathematics, United Kingdom, 2002.

[86] LewisResearchCenter. Centaur Space Vehicle Pressurized Propellant Feed

System Tests. Technical Note TN-D-6876, NASA, 1972.

[87] T. Y. Lin and D. Baker. Analysis and Testing of Propellant Feed System

Priming Process. Journal of Propulsion and Power, Vol. 11(No.3):pp. 505 –

512, May - June 1995.

[88] P.J. Linstrom and W.G. Mallard. NIST Chemistry WebBook, NIST Stan-

dard Reference Database Number 69. National Institute of Standards and

Technology, Gaithersburg MD, 20899, 2011. http://webbook.nist.gov, (re-

trieved July 14, 2011).

ix

Bibliography

[89] R. W. Lockhart and R. C. Martinelli. Proposed Correlation of Data for

Isothermal Two Phase Flow, Two Component Flow in Pipes. Chem. Eng.

Prog., Vol. 45:pp. 38 – 49, 1949.

[90] P. C. Lozano-Tovar. Dynamic Models for Liquid Rocket Engines with

Health Monitoring Application. Master’s thesis, Massachusetts Institute of

Technology, 1998.

[91] C. ManWetti. Transient Behaviour Modelling of Liquid Rocket Engine

Components. PhD thesis, Maschinenwesen der Rheinisch-Westfälischen

Technischen Hochschule Aachen, 2009.

[92] C. ManWetti. Start-Up Transient Simulation of a Pressure Fed LOX/LH2

Upper Stage Engine Using the Lumped Parameter-based MOLIERE Code. In

46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit,

number AIAA 2010-7046, 2010.

[93] C. ManWetti. Water Hammer Simulation Using a Lumped Parameter Model.

In IAF, editor, Space Propulsion Conference, 2010.

[94] D. Manski, C. Goertz, H. Saßnick, J. R. Hulka, B. D. Goracke, and D. J. H.

Levack. Cycles for Earth-to-Orbit Propulsion. Journal of Propulsion and

Power, Vol. 14(No.5):pp. 588–604, September-October 1998.

[95] N. B. McNelis and M. S. Haberbusch. Hot Fire Ignition Test with DensiVed

Liquid Hydrogen Using a RL10B-2 Cryogenic H2/O2 Rocket engine. In

33rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit,

number AIAA 1997-2688, 1997.

[96] M. Meyer. Electrically Heated Tube Investigation of Cooling Channel Geom-

etry EUects. In 31st ASME, SAE, and ASEE, Joint Propulsion Conference


[97] NASA. Liquid Rocket Engine Centrifugal Flow Turbopumps. Technical

Report SP-8109, NASA, December 1973.

[98] NASA. Turbopump Systems for Liquid Rocket Engines. Technical Report

SP-8107, NASA, August 1974.

x

Bibliography

[99] J. Nathman, J. Niehaus, J. C. Sturgis, A. Le, and J. Yi. Preliminary

Study of Heat Transfer Correlation Development and Pressure Loss

Behavior in Curved High Aspect Ratio Coolant Channels. In 44th


ber AIAA 2008-5240, 2008.

[100] D. Nguyen and A. Martinez. Versatile Engine Design Software. In 28th


ber AIAA 1993-2164, 1993.

[101] G. Odonneau, G. Albano, V. Leuduere, and J. Masse. Carins: A New

Versatile and Flexible Tool for Engine Transient Prediction - development

status. In 6th International Symposium on Launcher Technologies, 2005.

[102] G. Odonneau, G. Albano, and J. Masse. Carins: A New Versatile and Flexible

Tool for Engine Transient Prediction. In 4th International Conference on

Launcher Technology, Space Launcher Liquid Propulsion, 2002.

[103] A. J. Pavli, J. K. Curley, P. A. Masters, and R. M. Schwartz. Design and

Cooling Performance of a Dump Cooled Rocket Engine. Technical Note TN

D-3532, NASA, August 1966.

[104] S. Peery and A. Minick. Design and Development of an Advanced Expander

Combustor. In 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference


[105] P. Pempie and L. Boccaletto. LOX/CH4 Expander Upper Stage Engine. In

IAF, editor, 55th International Astronautical Congress, 2004.

[106] M. Pizzarelli, F. Nasuti, and M. Onofri. A SimpliVed Model for the Analysis

of Thermal StratiVcation in Cooling Channels. In EUCASS, 2nd European

Conference for Aerospace Sciences, July 2007.

[107] M. Pizzarelli, F. Nasuti, and M. Onofri. Flow Analysis of

Transcritical Methane in Rectangular Cooling Channels. In 44th


ber AIAA 2008-4556, July 2008.

xi

Bibliography

[108] M. Pizzarelli, F. Nasuti, and M. Onofri. Investigation of Transcriti-

cal Methane Flow and Heat Transfer in Curved Cooling Channels. In

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, number AIAA

2009-5304, August 2009.

[109] M. Pizzarelli, F. Nasuti, R. Paciorri, and M. Onofri. Numerical Analysis of

Three-Dimensional Flow of Supercritical Fluid in Asymmetrically Heated

Channels. AIAA Journal, 47(11):pp. 2534–2543, November 2009.

[110] Pratt&Whitney. Design Report For RL10A-3-3 Rocket Engine. Contractor

Report CR-80920, NASA, 1966.

[111] Pratt&Whitney. Rl10a-3-3 rocket engine oxidizer pump development pro-

gram. Contract Report CR-83547, NASA, 1966.

[112] Pratt&Whitney. Design Study of RL-10 Derivatives - Final Report. Technical

Report CR-120145, NASA, 1973.

[113] Pratt&Whitney. Design and Analysis Report for the RL10-IIB Breadboard

Low Thrust Engine. Contract Report NASA-CR-174857, NASA, 1984.

[114] Pratt&Whitney. RL10 ignition limits test for Shuttle Centaur. Contract

Report NASA-CR-183199, NASA, 1987.

[115] Pratt&Whitney. Shuttle Centaur Engine Cooldown Evaluation and EUects

of Expanded Inlets on Start Transient. Contract Report NASA-CR-183198,

NASA, 1987.

[116] D. Preclik, D. Wiedmann, W. Oechslein, and J. Kretschmer. Cryogenic

Calorimeter Chamber Experiments and Heat Transfer Simulations. In



[117] D. C. Pytanowski. Rocket-Engine Control-System Reliability-Enhancement

Analysis. In IEEE, editor, Annual RELIABILITY and MAINTAINABIL-

ITY Symposium, 1999.

xii

Bibliography

[118] V. Rachuk and N. Titkov. The First Russian LOX-LH2 Expander Cycle LRE:

RD0146. In 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference,


[119] D. Ramesh and M. Aminpoor. Nonlinear, Dynamic Simulation of an

Open Cycle Liquid Rocket Engine. In 43rd AIAA/ASME/SAE/ASEE Joint

Propulsion Conference and Exhibit, number AIAA 2007-5507, 2007.

[120] R. Reid, J.M Prausnitz, and B. Poling. The Properties of Gases and Liquids.

McGraw-Hill, 1987.

[121] R. Rhote-Vaney, V. Thomas, and A. Lekeux. Transient Modeling of Cryo-

genic Rocket Engines a Modular Approach. In 4th International Confer-

ence on Launcher Technology, Space Launcher Liquid Propulsion, 2002.

[122] José Ramón Alarcón Rodríguez. LOX/H2 engine heat transfer, equilibrium

and heat transfer analysis program. Technical report, ESA-ESTEC, TEC-

MPC, 2001. Draft version.

[123] P.L. Roe. Approximate Riemann solvers, parameter vectors, and diUerence

schemes. J. Comput. Phys., 43:357–372, 1981.

[124] E. K. Ruth, H. Ahn, R. L. Baker, and M. A. Brosmer. Advanced Liquid

Rocket Engine Transient Model. In AIAA/ASME/SAE/ASEE 26th Joint

Propulsion Conference and Exhibit, 1990.

[125] G. Saßnick, H. D.and Krülle. Numerical simulation of transients in feed

systems for cryogenic rocket engines . In ASME, SAE, and ASEE, Joint

Propulsion Conference and Exhibit, 31st, number AIAA 1995-2967, 1995.

[126] G. Schmidt, M. Popp, and T. Fröhlich. Design Studies for a 10 Ton Class High

Performance Expander Cycle Engine. In 34th AIAA/ASME/SAE/ASEE

Joint Propulsion Conference and Exhibit, number AIAA 1998-3673, 1998.

[127] R. SchuU, M. Maier, O. Sindiy, C. Ulrich, and S. Fugger. Integrated Modeling

& Analysis for a LOX/Methane Expander Cycle Engine Focusing on Re-

generative Cooling Jacket Design. In 42nd AIAA/ASME/SAE/ASEE Joint

Propulsion Conference, number AIAA 2006-4534, July 2006.

xiii

Bibliography

[128] P. F. Seitz and R. F. Searle. Space Shuttle Main Engine Control System.

In National Aerospace Engineering and Manufacturing Meeting, number

Paper 740927. Society of Automotive Engineers, October 1973.

[129] R. Sekita, A. Watanabel, K. Hirata, and T. Imoto. Lessons Learned from H-2

Failure and Enhancement of H-2A Project. Acta Astronautica, Vol. 48:pp.

431 – 438, 2001.

[130] A. J. StepanoU. Centrifugal and Axial Flow Pumps, 2nd edition. J. Wiley

and Sons, 1957.

[131] D. Suslov and M. Oschwald. Testfälle zur Validierung von Software für die

Berechnung des Wärmeübergangs in regeneraativ gekühlten Schbkammern.

Technical Report DLR-LA - WT-DO-019, DLR, 2010.

[132] D. Suslov, A. Woschnak, J. Sender, and M. Oschwald. Test specimen design

and measurement technique for investigation of heat transfer processes

in cooling channels of rocket engines under real thermal conditions. In



[133] George Paul Sutton and Oscar Biblarz. Rocket propulsion elements. John

Wiley & Sons, New York, NY, USA, 7th edition, 2001.

[134] A. Tarafder and S. Sarangi. CRESP-LP - A Dynamic Simulator for Liquid-

Propellant Rocket Engines. In 36th AIAA/ASME/SAE/ASEE Joint Propul-

sion Conference and Exhibit, number AIAA 2000-3768, 2000.

[135] J. R. Thome. Engineering Data Book III. Wolverine Tube. Inc, 2010.

[136] A. P. Tishin and L. P. Gurova. Liquid Rocket Engine Modeling. Proceedings

of the Aircraft Engineering College, Vol. 32(3):pp. 99 – 101, 1989.

[137] T. M. Tomsik. A Hydrogen-Oxygen Rocket Engine Coolant Passage Design

Program (RECOP) for Fluid-cooled Thrust Chambers and Nozzles. Technical

Note NASA-N95-70893, NASA, 1994.

xiv

Bibliography

[138] E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics:

a Practical Introduction. Springer, 1999.

[139] E. Turkel. Preconditioned methods for solving the incompressible and low

speed compressible equations. J. Comput. Phys., 72:277–298, 1987.

[140] A. Vaidyanathan, J. Gustavsoon, and C. Segal. One- and Three-Dimensional

Wall Heat Flux Calculations in a O2/H2 System. Journal of Propulsion and

Power, Vol. 26(No.1):pp. 186 – 188, January - February 2010.

[141] W. M. Van Lerberghe, J. L. Emdee, and R. R. Foust. Enhanced Reliability

Features of the RL10E-1 Engine. Acta Astronautica, Vol. 41(No.4):pp. 197 –

207, 1997.

[142] J. P. Veres. Centrifugal and Axial Pump Design and OU-Design Performance

Prediction. Technical Memorandum NASA-TM-106745, NASA, 1994.

[143] J. P. Veres and T. M. Lavelle. Mean Line Pump Flow Model in Rocket Engine

System Simulation. Technical MemorandumNASA-TM-2000-210574, NASA,

November 2000.

[144] G. Volpe. Performance of Compressible Flow Codes at Low Mach Number.

AIAA, 31(1):49–56, 1993.

[145] M. F. Wadel. Comparison of High Aspect Ratio Cooling Channels Designs

for a Rocket Combustion Chamber with Development of an Optimized

Design. Technical Memorandum NASA-TM-1998-206313, NASA, January

1998.

[146] M. F. Wadel and M. L. Meyer. Validation of High Aspect Ratio Cooling in a

89 kN (20,000 lb) Thrust Combustion Chamber. Technical Memorandum

NASA-TM-107270, NASA, 1996.

[147] G. B. Wallis. One-Dimensional Two Phase Flow. McGraw-Hill, 1969.

[148] D. W. Way and J. R. Olds. SCORES: Developing an Object-Oriented Rocket

Propulsion Analysis Tool. In 34th AIAA/ASME/SAE/ASEE Joint Propul-

sion Conference & Exhibit, number AIAA 1998-3227, 1998.

xv

Bibliography

[149] Wikipedia. Space Shuttle Main Engine.

http://en.wikipedia.org/wiki/Space_Shuttle_Main_Engine. retrieved

28/12/2011.

[150] A. Woschnak and M. Oschwald. Thermo- and Fluidmechanical Analysis

of High Aspect Ratio Cooling Channels. In AIAA/ASME/SAE/ASEE 37th

Joint Propulsion Conference and Exhibit, number AIAA 2001-3404, July

2001.

[151] A. Woschnak, D. Suslov, and M. Oschwald. Experimental and Numerical

Investigations of Thermal StratiVcation EUects. InAIAA/ASME/SAE/ASEE

39th Joint Propulsion Conference and Exhibit, number AIAA 2003-4615,

July 2003.

[152] N. Yamanishi, T. Kimura, M. Takahashi, K. Okita, and H. Negishi. Transient

Analysis of the LE-7A Rocket Engine Using the Rocket Engine Dynamic

Simulator (REDS). In AIAA/ASME/SAE/ASEE 40th Joint Propulsion Con-

ference and Exhibit, number AIAA 2004-3850, July 2004.

[153] Yanzhong, Cui, Chen, and Yuanyuan. Pressure wave propagation character-

istics in a two-phase Wow pipeline for liquid-propellant rocket. Aerospace

Science and Technology, 15:453–464, September 2011.

xvi

PhD Thesis FDM

Documents

Transcript of PhD Thesis FDM