modelling and simulation of liquid rocket...
TRANSCRIPT
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
6.14. Transient results - part 2 . . . . . . . . . . . . . . . . . . . . . . . . 172
6.15. Transient results - part 3 . . . . . . . . . . . . . . . . . . . . . . . . 173
6.16. Transient results - part 4 . . . . . . . . . . . . . . . . . . . . . . . . 174
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
6.20. Shut-down results - part 2 . . . . . . . . . . . . . . . . . . . . . . . 181
6.21. Shut-down results - part 3 . . . . . . . . . . . . . . . . . . . . . . . 182
6.22. Shut-down results - part 4 . . . . . . . . . . . . . . . . . . . . . . . 183
6.23. TCV throttle results - part 1 . . . . . . . . . . . . . . . . . . . . . . 187
6.24. TCV throttle results - part 2 . . . . . . . . . . . . . . . . . . . . . . 188
6.25. TCV throttle results - part 3 . . . . . . . . . . . . . . . . . . . . . . 189
6.26. TCV throttle results - part 4 . . . . . . . . . . . . . . . . . . . . . . 190
6.27. OCV throttle results - part 1 . . . . . . . . . . . . . . . . . . . . . . 193
6.28. OCV throttle results - part 2 . . . . . . . . . . . . . . . . . . . . . . 194
6.29. OCV throttle results - part 3 . . . . . . . . . . . . . . . . . . . . . . 195
6.30. OCV throttle results - part 4 . . . . . . . . . . . . . . . . . . . . . . 196
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
3. ESPSS: European Space Propulsion System Simulation
• 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
3. ESPSS: European Space Propulsion System Simulation
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
3. ESPSS: European Space Propulsion System Simulation
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
3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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)
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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-
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3. ESPSS: European Space Propulsion System Simulation
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),
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3. ESPSS: European Space Propulsion System Simulation
• 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
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3. ESPSS: European Space Propulsion System Simulation
• 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
calculated as follows:
ρgas = ρvap + ρnc
Cpgas = (ρvapCpvap + ρncCpnc)/ρgas
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
• 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.
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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:
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3. ESPSS: European Space Propulsion System Simulation
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,
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
• 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.
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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)
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3. ESPSS: European Space Propulsion System Simulation
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
3. ESPSS: European Space Propulsion System Simulation
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:
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3. ESPSS: European Space Propulsion System Simulation
- 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
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3. ESPSS: European Space Propulsion System Simulation
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:
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3. ESPSS: European Space Propulsion System Simulation
• 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
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
• 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
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3. ESPSS: European Space Propulsion System Simulation
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:
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3. ESPSS: European Space Propulsion System Simulation
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)
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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).
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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)
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3. ESPSS: European Space Propulsion System Simulation
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
calculated as follows:
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)
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
calculated as follows:
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
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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)
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3. ESPSS: European Space Propulsion System Simulation
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)
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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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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
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3. ESPSS: European Space Propulsion System Simulation
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.
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3. ESPSS: European Space Propulsion System Simulation
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)
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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
4. Steady State Library
Figure 4.1.: Components in the Steady State library
79
4. Steady State Library
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
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4. Steady State Library
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)
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4. Steady State Library
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.
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4. Steady State Library
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)
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4. Steady State Library
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 )
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4. Steady State Library
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
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4. Steady State Library
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.
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4. Steady State Library
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
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4. Steady State Library
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)
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4. Steady State Library
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
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4. Steady State Library
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
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4. Steady State Library
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,
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4. Steady State Library
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,
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4. Steady State Library
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.
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4. Steady State Library
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
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4. Steady State Library
• 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
4. Steady State Library
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.
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4. Steady State Library
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
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4. Steady State Library
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.
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4. Steady State Library
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
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4. Steady State Library
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.
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4. Steady State Library
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
4. Steady State Library
Figure 4.5.: Turbopump test case: HM7B power pack transient schematic
Figure 4.6.: Turbopump test case: HM7B power pack steady state schematic
102
4. Steady State Library
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
4. Steady State Library
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
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4. Steady State Library
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
4. Steady State Library
Figure 4.9.: HM7B engine system schematic
106
4. Steady State Library
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
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4. Steady State Library
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
4. Steady State Library
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.
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4. Steady State Library
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
4. Steady State Library
Figure 4.10.: Schematic of the RL-10 engine
111
4. Steady State Library
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
4. Steady State Library
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
5. Transient Modelling
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
5. Transient Modelling
(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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
• 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
5. Transient Modelling
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. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
(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
5. Transient Modelling
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
5. Transient Modelling
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
5. Transient Modelling
Figure 5.9.: Schematic of the experimental test case
136
5. Transient Modelling
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
5. Transient Modelling
(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
5. Transient Modelling
(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
(d) Thermocouples temperatures, x = 85 mm,pc = 58 bar
(e) Thermocouples temperatures,x = 119 mm,pc = 58 bar
(f) Thermocouples temperatures, x = 152 mm,pc = 58 bar
Figure 5.11.: Wall and Wuid thermal stratiVcation, AR = 9.2, pc = 58 bar
139
5. Transient Modelling
(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
(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 5.12.: Wall and Wuid thermal stratiVcation, AR = 30, pc = 88 bar
140
5. Transient Modelling
(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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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)
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6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
(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
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6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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)
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6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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].
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6. Integrated Validation: RL-10 design and analysis
(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
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6. Integrated Validation: RL-10 design and analysis
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.
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6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
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6. Integrated Validation: RL-10 design and analysis
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].
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6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
Figure 6.10.: RL-10A-3-3A schematic model
164
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
Figure 6.11.: RL-10A-3-3A Valve schedule for Start-up Simulation [15]
166
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
= 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
6. Integrated Validation: RL-10 design and analysis
(a) Chamber Pressure
(b) LOX Pump Shaft Speed
Figure 6.13.: Transient results - part 1
171
6. Integrated Validation: RL-10 design and analysis
(a) LOX Pump Discharge Pressure
(b) LOX Pump Inlet Pressure
Figure 6.14.: Transient results - part 2
172
6. Integrated Validation: RL-10 design and analysis
(a) LOX Engine inlet mass Wow
(b) Fuel Engine inlet mass Wow
Figure 6.15.: Transient results - part 3
173
6. Integrated Validation: RL-10 design and analysis
(a) Venturi inlet Pressure
(b) Turbine Inlet Temperature
Figure 6.16.: Transient results - part 4
174
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
Figure 6.17.: RL-10A-3-3A Valve schedule for Shut-down Simulation [15]
176
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
(a) Chamber Pressure
(b) LOX Pump Shaft Speed
Figure 6.19.: Shut-down results - part 1
180
6. Integrated Validation: RL-10 design and analysis
(a) LOX Pump Discharge Pressure
(b) LOX Pump Inlet Pressure
Figure 6.20.: Shut-down results - part 2181
6. Integrated Validation: RL-10 design and analysis
(a) LOX Engine inlet mass Wow
(b) Fuel Engine inlet mass Wow
Figure 6.21.: Shut-down results - part 3182
6. Integrated Validation: RL-10 design and analysis
(a) Venturi inlet Pressure
(b) Fuel Pump Inlet Pressure
Figure 6.22.: Shut-down results - part 4
183
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
(a) Chamber Pressure
(b) Chamber Mixture Ratio
Figure 6.23.: TCV throttle results - part 1
187
6. Integrated Validation: RL-10 design and analysis
(a) LOX Pump Discharge Pressure
(b) Fuel Pump Discharge Pressure
Figure 6.24.: TCV throttle results - part 2
188
6. Integrated Validation: RL-10 design and analysis
(a) LOX chamber inlet mass Wow
(b) Fuel chamber inlet mass Wow
Figure 6.25.: TCV throttle results - part 3
189
6. Integrated Validation: RL-10 design and analysis
(a) Turbine Mass Flow
(b) Turbine Shaft Speed
Figure 6.26.: TCV throttle results - part 4
190
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
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
6. Integrated Validation: RL-10 design and analysis
(a) Chamber Pressure
(b) Chamber Mixture Ratio
Figure 6.27.: OCV throttle results - part 1
193
6. Integrated Validation: RL-10 design and analysis
(a) LOX Pump Discharge Pressure
(b) Fuel Pump Discharge Pressure
Figure 6.28.: OCV throttle results - part 2
194
6. Integrated Validation: RL-10 design and analysis
(a) LOX chamber inlet mass Wow
(b) Fuel chamber inlet mass Wow
Figure 6.29.: OCV throttle results - part 3
195
6. Integrated Validation: RL-10 design and analysis
(a) Turbine Mass Flow
(b) Turbine Shaft Speed
Figure 6.30.: OCV throttle results - part 4
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
A. Implementation of Up-wind Roe Scheme
• 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. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
A. Implementation of Up-wind Roe Scheme
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
B. Friction Factor Correlations
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
B. Friction Factor Correlations
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
B. Friction Factor Correlations
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
C. Film CoeXcient Calculation
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
C. Film CoeXcient Calculation
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.
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