Finite Element Modeling and Simulation of the Effect of Water Injection on
Gas Turbine Combustor NOx Emissions and Component Temperature
Joseph M. Basile
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Dr. Sudhangshu Bose, Thesis Adviser
_________________________________________
Dr. Ernesto Gutierrez-Miravete, Committee Member
_________________________________________
Dr. Norberto Lemcoff, Committee Member
Rensselaer Polytechnic Institute
Troy, New York
November 2014
(For Graduation December 2014)
© Copyright 2014
by
Joseph M. Basile
All Rights Reserved
ii
CONTENTS
NOMENCLATURE .......................................................................................................................v
LIST OF TABLES ...................................................................................................................... vii
LIST OF FIGURES .................................................................................................................... viii
ACKNOWLEDGMENT ................................................................................................................x
ABSTRACT ................................................................................................................................. xi
1.
INTRODUCTION ........................................................................................................1
1.1. Gas Turbine Operation ................................................................................................1
1.2. Nitrogen Oxide Formation ...........................................................................................3
1.2.1. Thermal Nitric Oxide .......................................................................................4
1.2.2. Prompt Nitric Oxide .........................................................................................4
1.2.3. Nitrous Oxide Mechanism................................................................................5
1.2.4. Fuel Nitrous Oxide ............................................................................................6
1.3. Water Injection .............................................................................................................6
2.
METHODOLOGY .......................................................................................................7
2.1. Theory ............................................................................................................................7
2.1.1. Fluid Flow and Momentum .............................................................................8
2.1.2. Energy Balance .................................................................................................9
2.1.3. Equation of State.............................................................................................10
2.1.4. Turbulence.......................................................................................................10
2.1.5. Discretization of Model Equations ................................................................11
2.2. Modeling ......................................................................................................................11
2.2.1. Baseline Selection ............................................................................................11
2.2.2. Combustor Modeling ......................................................................................12
2.2.3. Prediction of NOx Emissions ..........................................................................21
2.2.4. 2D Axisymmetric Simulation of Combustion Process .................................22
2.2.5. 2D Axisymmetric Simulation of Water Injection ........................................27
iii
2.2.6. 3D Simulation of Combustion Process ..........................................................35
2.2.7. 3D Simulation of Water Injection .................................................................40
3.
RESULTS AND DISCUSSION .................................................................................43
3.1. 2D Axisymmetric Models ...........................................................................................43
3.2. 3D Models ....................................................................................................................47
4.
CONCLUSIONS .........................................................................................................52
5.
REFERENCES............................................................................................................53
APPENDIX A: Determination of Boundary Conditions ........................................................54
Appendix A-1: 2D axisymmetric model, no water injection ............................................54
Appendix A-2: 2D axisymmetric model, with water injection .........................................55
Appendix A-3: 3D model, no water injection.....................................................................58
Appendix A-4: 3D model, with water injection .................................................................60
iv
NOMENCLATURE
∇
a
Cp
CR
𝛿𝑖𝑗
𝜕
𝜕𝑡
𝜕
𝜕𝑛
𝐷
𝐷𝑡
𝐸𝑡
𝛾
e
ε
f
f
𝐟𝐢
FA
ℎ𝑓̇
𝐻𝑓
Hw
i
J
k
k
K
kg
N
N2O
NO
NO2
NOx
m
𝑚̇
𝑝
ρ
Pa
∏ij
𝒒
Gradient
Ambient
Heat capacity at constant pressure (J/kg)
Compression ratio
Kronecker delta
Partial derivative with respect to time
Partial derivative with respect to some variable n
Derivative with respect to time
Total energy per unit volume (J/m3)
Ratio of specific heats (1.4)
Natural exponent (~2.71)
Turbulent dissipation rate (J/kg/s)
Fuel
Body force per unit mass (N)
Body force per unit mass in the i direction (N)
Fuel to air ratio
Heat of combustion (W)
Heating value of fuel (J/kg)
Latent heat of evaporation for water (J/kg)
Problem domain inlet
Joules (kg*m2/s2)
Thermal conductivity (W/m/K)
Turbulent kinetic energy (J/kg)
Degrees Kelvin
Kilograms
Newtons (kg*m/s2)
Nitrous oxide
Nitric oxide
Nitrogen dioxide
Nitrogen oxides
Meters
Mass flow rate (kg/s)
Pressure (Pa)
Density (kg/m3)
Pascals (N/m2)
Stress tensor (Pa)
Heat flux vector (J/s)
v
𝑄
R
s
T
𝐮
𝑢
ui
𝜇
𝜇𝑇
𝑣
w
𝑤
W
External heat addition per unit volume (J/m3)
Specific gas constant for air (287 J/kg/K)
Seconds
Temperature (K)
Velocity Vector (m/s)
Velocity magnitude in the x direction (m/s)
Velocity component in the i direction (m/s)
Viscosity
Turbulent viscosity (Pa*s)
Velocity magnitude in the y direction (m/s)
water
Velocity magnitude in the z direction (m/s)
Watts (J/s)
vi
LIST OF TABLES
Table 1: Values used for independent turbulence parameters ..................................................... 11
Table 2: Engine Performance: Key Parameters [9] ..................................................................... 18
Table 3: Problem domain inlet boundary conditions and target flow rates ................................. 21
Table 4: NOx emissions dependence on combustor exit temperature [10] ................................. 21
Table 5: Operating characteristics for the combustor, without water injection .......................... 27
Table 6: Operating characteristics for the combustor, with water injection ............................... 34
Table 7: Target flow rates and boundary conditions, no water injection .................................... 37
Table 8: Operating characteristics for the 3D combustor model, without water injection ......... 38
Table 9: 3D model target flow rates and boundary conditions, with water injection ................. 41
Table 10: Operating characteristics for the 3D combustor model, without water injection........ 42
Table 11: Operating characteristics - 2D axisymmetric models ................................................. 46
Table 12: 2D NOx production with and without water injection ................................................ 46
Table 13: Operating characteristics - 3D models ........................................................................ 50
Table 14: 3D NOx production with and without water injection ................................................ 50
Table 15: 2D model iterative convergence toward target mass flow rates, no water injection ... 54
Table 16: 2D model iterative convergence toward target mass flow rates, with water injection 55
Table 17: 3D model iterative convergence toward target mass flow rates, no water injection ... 58
Table 18: 3D model iterative convergence toward target mass flow rates, with water injection 60
vii
LIST OF FIGURES
Figure 2: Section view showing the four thermodynamic states in a gas turbine engine [2] ........ 2
Figure 3: Brayton Cycle P-V Diagram .......................................................................................... 3
Figure 4: Combustion Chamber Dimensional Analysis [9] ........................................................ 13
Figure 5: Full 3D combustor model ............................................................................................ 13
Figure 6: Combustion Chamber Simplified Model - Isometric View ......................................... 14
Figure 7: Combustion Chamber Model - Side View ................................................................... 15
Figure 8: Engine Air Flow Region .............................................................................................. 16
Figure 9: Air flow region and combustion chamber in COMSOL.............................................. 17
Figure 10: 2D Axisymmetric air flow region and combustion chamber in COMSOL ............... 18
Figure 11: 2D axisymmetric combustor model with combustion region .................................... 23
Figure 12: 2D axisymmetric mesh, no water injection ............................................................... 24
Figure 13: Air mass flow rate dependence on backpressure with quadratic curve fit ................. 25
Figure 14: Fuel mass flow rate dependence on fuel inlet pressure with linear curve fit ............. 26
Figure 15: 2D axisymmetric model temperature distribution, no water injection ...................... 27
Figure 17: 2D axisymmetric mesh, with water injection ............................................................ 30
Figure 18: Compressor air flow rate as a function of combustor backpressure .......................... 31
Figure 19: Fuel mass flow rate as a function of fuel inlet pressure............................................. 32
Figure 20: Water mass flow rate as a function of water inlet pressure ....................................... 33
Figure 21: 2D axisymmetric model temperature distribution, with water injection ................... 34
Figure 22: 3D combustion chamber model with combustion region .......................................... 35
Figure 23: Fuel flow from the fuel injector through the combustion region ............................... 36
Figure 24: 3D problem domain mesh, no water injection ........................................................... 37
Figure 25: 3D streamlines showing flow from the compressor outlet to combustor outlet, no
water injection ............................................................................................................................. 38
Figure 26: Temperature profile of flow through the domain, no water injection ....................... 39
Figure 27: 3D combustion chamber model with heat absorbing region ..................................... 40
Figure 28: 3D problem domain mesh, with water injection ........................................................ 41
Figure 29: 3D streamlines showing flow from the compressor outlet and water inlet to
combustor outlet .......................................................................................................................... 42
Figure 30: Temperature profile of flow through the domain, with water injection .................... 43
viii
Figure 31: 2D absolute pressure results without water injection (left) and with (right) ............. 44
Figure 32: 2D Velocity magnitude results without water injection (left) and with (right) ......... 44
Figure 33: 2D Temperature results without water injection (left) and with (right) .................... 45
Figure 34: 2D wall temperature without water injection (left) and with (right) ......................... 47
Figure 35: 3D absolute pressure results without water injection (left) and with (right) ............. 48
Figure 36: 3D Velocity magnitude results without water injection (left) and with (right) ......... 48
Figure 37: 3D Temperature results without water injection (left) and with (right) .................... 49
Figure 38: 3D wall temperature without water injection (left) and with (right) ......................... 51
Figure 39: 2D model curve fit and predicted backpressure value to achieve the target inlet air
mass flow rate, no water injection ............................................................................................... 54
Figure 40: 2D model curve fit and predicted fuel line pressure value to achieve the target fuel
mass flow rate, no water injection ............................................................................................... 55
Figure 41: 2D model curve fit and predicted backpressure value to achieve the target inlet air
mass flow rate, with water injection ............................................................................................ 56
Figure 42: 2D model curve fit and predicted fuel line pressure value to achieve the target inlet
air mass flow rate, with water injection ...................................................................................... 57
Figure 43: 2D model curve fit and predicted water injection pressure value to achieve the target
inlet air mass flow rate ................................................................................................................ 58
Figure 44: 3D model curve fit and predicted backpressure value to achieve the target inlet air
mass flow rate, no water injection ............................................................................................... 59
Figure 45: 3D model curve fit and predicted fuel line pressure value to achieve the target fuel
mass flow rate, no water injection ............................................................................................... 59
Figure 46: 3D model curve fit and predicted backpressure value to achieve the target inlet air
mass flow rate, with water injection ............................................................................................ 60
Figure 47: 3D model curve fit and predicted fuel line pressure value to achieve the target inlet
air mass flow rate, with water injection ...................................................................................... 61
Figure 48: 3D model curve fit and predicted water injection pressure value to achieve the target
inlet air mass flow rate ................................................................................................................ 62
ix
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
x
ABSTRACT
Gas turbine engines are airbreathing engines which use the expansion of high temperature
exhaust through an axial turbine to generate shaft work and, if the application requires it, thrust.
Applications include electrical generating equipment and aircraft engines. These engines
typically operate at temperatures high enough to spontaneously generate nitrogen oxides
through various chemical mechanisms. These combustion byproducts are considered a
significant source of environmental pollution and reduction of such emissions is a topic of
contemporary research.
In addition to pollution, the high temperature operation of these engines also necessitates the
use of heat resistant materials for engine components. High temperatures can damage or wear
engine components, increasing maintenance costs.
Several numerical models have been developed to show that the production of nitrogen oxides
can be reduced while simultaneously protecting engine components. The models and
simulations used show that the surfaces of internal engine components can be maintained at low
temperatures by spraying atomized liquid water into the flow stream. The temperature is
reduced because heat from the combustion process will be absorbed by the water as it
evaporates. The addition of water will also increase the overall mass flow rate through the
engine and reduce the resultant flame temperature, thereby lowering the emission of nitrogen
oxides. By injecting water into an appropriate part of the engine, these benefits can be realized
simultaneously, thus increasing the advantage to implementing a water injection system.
A 2D axisymmetric model of a gas turbine interior and combustion chamber showed general
correlation to available test data and demonstrated the feasibility of the concept. A 3D model
considering a 30° segment of the engine interior and combustion chamber showed improved
accuracy with respect to available test data and clearly demonstrated the benefits of careful
selection of water injection location. While a 2D axisymmetric model can provide general
guidance for design decisions, a full 3D model is necessary for accurately predicting the best
injection site.
xi
1.
INTRODUCTION
1.1.
Gas Turbine Operation
Gas turbine engines are airbreathing engines which use the expansion of high temperature
exhaust through an axial turbine to generate shaft work and, if the application requires it, thrust.
Typically, an axial compressor stage at the intake raises the temperature and pressure of
incoming working fluid (typically air). From the compressor it enters the combustion chamber,
where gaseous or atomized fuel is injected. The combustion process is typically started with an
independent ignition source, but is self sustaining with adequate intake air and fuel flow rates.
From the combustion chamber it flows through a turbine stage where work is extracted and then
is expanded out the engine exhaust. Since their initial development in the early and mid
twentieth century, gas turbine engines have evolved into a high power, efficient means of
producing power and thrust. They have been used in a wide variety of applications, most visibly
as the primary power source for commercial aviation aircraft. [1] A typical gas turbine engine
cutaway, a General Electric J85-GE-17A turbojet, is shown in Figure 1.
Figure 1: Sectioned Turbojet Gas Turbine Engine (air flow is left to right) [2]
1
More specifically, gas turbine engines operate based on an open loop Brayton Cycle. There
are four different thermodynamic processes that occur between the four distinct states in the
cycle. The different regions to which these states correspond are shown in Figure 2.
Figure 2: Section view showing the four thermodynamic states in a gas turbine engine [2]
The following is a description of the ideal Brayton Cycle, and ignores losses that would be
found in a real engine. State 1 is the ambient gas which enters the engine intake. Between states
1 to 2, the gas compressed isentropically by an axial-flow rotating compressor. Therefore, at
state 2 it is at elevated temperature and pressure.
From the compressor stage it enters the combustion chamber, where the burning fuel heats
it isobarically from states 2 to 3.
From states 3 to 4, the gas expands isentropically through the turbine. For a jet engine, work
is extracted as a torque on the turbine shaft and a reaction force (thrust) acting toward the
intake. The turbine shaft is directly coupled to the compressor and thus work extracted by the
turbine stage is used to compress the gas at the intake. This makes the cycle self-sustaining.
From states 4 to 1, the gas is cooled isobarically back to ambient conditions, typically outside of
2
the engine housing. Figure 3 shows a typical pressure vs. volume diagram for an ideal Brayton
Cycle.
3
Pressure (N/m2)
2
1
4
Volume (m3)
Figure 3: Brayton Cycle P-V Diagram
The thermodynamic efficiency of a real engine operating on the Brayton Cycle is dependent on
the temperature change from State 1 to State 2 during compression. This means that a higher
compression ratio will yield higher combustor inlet temperatures and, therefore, a higher cycle
efficiency.
1.2.
Nitrogen Oxide Formation
A side effect of the operation of gas turbine engines is the formation of pollutants in the
exhaust. Nitrogen oxides are one of the most harmful pollutants created during the combustion
process. Nitrogen oxides generally include nitric oxide (NO) and nitrogen dioxide (NO2), and
emissions generally refer to the sum total of NO and NO2 simply as NOx. Generally, high
3
combustion temperatures causes a reaction between the oxygen and nitrogen in the air flow and
combustion products, forming NOx. The formation of NOx increases as the compression ratio
and flame temperature of the engine increase.
Gas turbine development has had a clear trend toward higher compression ratios and
corresponding higher turbine inlet temperatures. This trend results in engines with increased
performance and efficiency, but has also led to increasing nitrogen oxide emissions. Nitrogen
oxides are formed when gaseous nitrogen from the intake oxidizes at high temperatures.
Nitrogen oxide formation increases rapidly as flame temperature increases, especially for
temperatures in excess of 1800K. [6] The U.S. government has identified nitrogen oxides as an
air pollutant and taken steps to control its emission from gas turbine engines. [3]
There are four distinct mechanisms through which NOx forms during the combustion process.
NOx can form through the thermal nitric oxide route, the prompt nitric oxide route, the nitrous
oxide (N2O) mechanism, and the fuel nitric oxide route. [4] [5]
1.2.1. Thermal Nitric Oxide
The thermal nitric oxide route forms NO based on the Zel'dovich mechanism. This mechanism
is described by the following reactions:
𝑂2 → 2𝑂
𝑂 + 𝑁2 → 𝑁𝑂 + 𝑁
𝑁 + 𝑂2 → 𝑁𝑂 + 𝑂
𝑁 + 𝑂𝐻 → 𝑁𝑂 + 𝐻
The first two reactions can be interpreted as a decomposition of the oxygen and nitrogen
molecules in air generating NOx at elevated temperatures. However, in gas turbine combustors,
the residence times of these components in the flame region are generally insufficient for the
flow to reach chemical equilibrium. [4]
1.2.2. Prompt Nitric Oxide
4
The prompt nitric oxide route occurs in the combustion region much faster than NOx formed by
the thermal NO route. This mechanism occurs when hydrocarbon radials react with N2 to form
hydrocyanic acid (HCN), which then reacts for form NO. This is described by the following
reaction:
𝑁2 + 𝐶𝐻 → 𝐻𝐶𝑁 + 𝑁
Although other reactions take place, this is the primary path for HCN, and therefore NO
formation. Formation of NO from HCN occurs according to the following reaction sequence:
𝐻𝐶𝑁 + 𝑂 → 𝑁𝐶𝑂 + 𝐻
𝑁𝐶𝑂 + 𝐻 → 𝑁𝐻 + 𝐶𝑂
𝑁𝐻 + 𝐻 → 𝑁 + 𝐻2
𝑁 + 𝑂𝐻 → 𝑁𝑂 + 𝐻
These reactions have a far lower activation energy than the thermal NO mechanism, and
therefore the prompt NO route is one of the primary means of NOx formation at temperatures
below 2000 K. [5]
1.2.3. Nitrous Oxide Mechanism
N2O forms from the reaction:
𝑁2 + 𝑂 = 𝑁2 𝑂
Another reaction which can occur during combustion is the formation of NO from N2O and O.
This takes place according to:
𝑁2 𝑂 + 𝑂 → 2𝑁𝑂
𝑁2 𝑂 + 𝐻 → 𝑁𝐻 + 𝑁𝑂
𝑁2 𝑂 + 𝐶𝑂 → 𝑁𝐶𝑂 + 𝑁𝑂
5
At fuel lean conditions, and at lower flame temperatures, the thermal and prompt formation
mechanisms are greatly reduced. The N2O mechanism is therefore is the primary mechanism
through which NOx forms at low combustion temperatures. [4]
1.2.4. Fuel Nitrous Oxide
Many hydrocarbon fuels contain fuel bound nitrogen, and oxidation of this fuel bound nitrogen
is another source of NOx emissions. These emissions are generally dependant on the fuel being
burned and are only loosely dependant on specific combustion conditions such as flame
temperature. [4]
Because fuel nitrous oxide is a direct result of the combustion of fuel, it is generally not feasible
to limit this source of NOx emissions. The other NOx formation mechanisms can be reduced by
lowering the flame temperature through injection of non-reactive fluid, often water, which
absorbs some of the heat of generated by the reaction.
1.3.
Water Injection
One of the most effective means of reducing NOx emissions from gas turbine engines is the
injection of water or steam into the flow stream. [3] This has the effect of the lowering flame
temperature by increasing the mass flow rate of fluid through the engine without increase the
heat generated by combustion. This effect is increased dramatically if liquid water is used, due
to the latent heat of evaporation as the water becomes steam within the engine.
Because the trend in higher gas turbine compression ratios and, therefore, increased NOx
generation is likely to continue, research into a reduction of nitrogen oxide formation in gas
turbine combustion chambers is warranted. Recent studies have shown that injecting atomized
water directly into the combustion chamber can potentially cut nitrogen oxide emissions by
roughly 50% on a commercial airliner during the takeoff/climbout phase of flight. [6]
Substantial research into the best method for water injection has been pursued. Studies have
considered the injection of atomized water into the compressor intake, the combustor intake, or
other various places within the air flow stream. [6] Mixing of water directly into the incoming
fuel has also been studied. [3] This thesis will describe a method to model and simulate the
6
injection of water into a gas turbine combustor to provide direct cooling for engine components
while simultaneously reducing NOx emissions.
2.
METHODOLOGY
The goal of this study was to create a numerical model of a gas turbine combustion chamber
and use it to simulate the effect of atomized liquid water on engine component temperature and
NOx emissions. This section explains the methodology for modeling and simulating the baseline
gas turbine combustion chamber within the interior of the engine. First, a baseline model was
created showing nominal operation. This was then compared to a second model simulating
water injection along the inner wall of the engine. This allowed a comparison of the operation
of the engine with and without water injection. This process was completed using a 2D
axisymmetric model to minimize processing time and then using a 3D model to maximize
accuracy.
The working hypothesis was that the injection of atomized water into the engine can be used to
protect engine components from excessive temperatures caused by fuel combustion while
simultaneously lowering NOx emissions. Such a strategy for water injection would be especially
useful in the aviation industry, as both NOx emissions and engine component temperatures are
highest during the takeoff and climbout phases of flight. [6]
The flow through the combustor, the combustion process, and water injection were modeled in
COMSOL 4.3. The software uses the principles of fluid dynamics to solve flow problems
numerically, a process called computational fluid dynamics (CFD), using finite element
discretization. The following sections describe the theory, the model, and the methodology used
to simulate the combustor in steady state operating conditions.
2.1.
Theory
The flow through the gas turbine combustor was modeled using CFD. CFD methods use a set of
partial differential equations (PDEs) to describe the properties of the fluids (and solids) in the
problem domain. These PDEs are solved to determine the values of the dependant variables.
7
In the combustion chamber models, the problem domain encompasses the interior of the engine
between the compressor outlet and turbine inlet, including the combustion chamber. Boundary
conditions establish the pressure and temperature of incoming fluids and the pressure at the
outlet. Conditions are also set within the problem domain to add or subtract heat at a
predetermined rate to simulate combustion products or phase changes. A set of PDEs is then
used to solve for the pressure, flow velocity, temperature, and other dependant variables within
the interior of the engine. The solution to these PDEs form a complete description of the
simulated fluid flow and heat fluxes through the interior of the problem domain.
2.1.1. Fluid Flow and Momentum
The flow of fluid through the problem domain is governed by the conservation of mass and the
conservation of fluid momentum. These principles are described by the following equations:
𝜕𝜌
+ ∇ ∙ (𝜌𝑉) = 0
𝜕𝑡
𝜌
𝐷𝑽
𝜕
𝜕𝑢𝑖 𝜕𝑢𝑗
2
𝜕𝜇𝑘
= 𝜌𝐟 − 𝛁𝑝 +
[𝜇 (
+
) − 𝛿𝑖𝑗 𝜇
]
𝐷𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗 𝜕𝑥𝑖
3
𝜕𝑥𝑘
(1)
(2)
Equation 1 is the continuity equation. The continuity equation ensures that mass is conserved
through the flow domain. It describes the rate of increase in density in a differential control
volume and the rate of mass flux passing into or out of a differential control volume. When
applied to a flow domain, a steady state solution will always show that the net mass flux across
the boundaries of the system is zero. [7]
Equation 2 is the Navier-Stokes equation. The Navier-Stokes equation describes the motion of
viscous, compressible, Newtonian fluids. In Cartesian coordinates, the Navier-Stokes equations
are:
(3)
8
𝜌
𝐷𝒖
𝜕𝑝 𝜕 2
𝜕𝑢 𝜕𝑣 𝜕𝑤
𝜕
𝜕𝑢 𝜕𝑣
𝜕
𝜕𝑤 𝜕𝑢
= 𝜌𝐟𝐱 −
+
[ 𝜇 (2
−
−
)] +
[𝜇 ( + )] + [𝜇 (
+ )]
𝐷𝑡
𝜕𝑥 𝜕𝑥 3
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝑦
𝜕𝑦 𝜕𝑥
𝜕𝑧
𝜕𝑥 𝜕𝑧
𝜌
𝐷𝒗
𝜕𝑝 𝜕
𝜕𝑣 𝜕𝑢
𝜕 2
𝜕𝑣 𝜕𝑢 𝜕𝑤
𝜕
𝜕𝑣 𝜕𝑤
= 𝜌𝐟𝐲 −
+
[𝜇 ( + )] +
[ 𝜇 (2
−
−
)] + [𝜇 ( +
)] (4)
𝐷𝑡
𝜕𝑦 𝜕𝑥
𝜕𝑥 𝜕𝑦
𝜕𝑦 3
𝜕𝑦 𝜕𝑥 𝜕𝑧
𝜕𝑧
𝜕𝑧 𝜕𝑦
𝜌
𝐷𝒘
𝜕𝑝 𝜕
𝜕𝑤 𝜕𝑢
𝜕
𝜕𝑣 𝜕𝑤
𝜕 2
𝜕𝑤 𝜕𝑢 𝜕𝑣
= 𝜌𝐟𝐳 −
+
[𝜇 (
+ )] +
[𝜇 ( +
)] + [ 𝜇 (2
−
− )] (5)
𝐷𝑡
𝜕𝑧 𝜕𝑥
𝜕𝑥 𝜕𝑧
𝜕𝑦
𝜕𝑧 𝜕𝑦
𝜕𝑧 3
𝜕𝑧 𝜕𝑥 𝜕𝑦
Equations 3 through 5 described the motion of viscous, compressible, laminar Newtonian fluids
in each of the 3 principle directions. They take into account viscous stresses and fluid pressure
acting on a differential control volume. [7]
2.1.2. Energy Balance
The first law of thermodynamics, also known as the law of conservation of energy, can be
applied to a differential control volume to describe the internal energy through the flow domain.
𝜕𝐸𝑡
𝜕𝑄
+ 𝛁 ∙ 𝐸𝑡 𝐮 =
− 𝛁 ∙ 𝒒 + 𝜌𝐟 ∙ 𝐮 + 𝛁(∏𝐢𝐣 ∙ 𝐮)
𝜕𝑡
𝜕𝑡
(6)
Equation 6 described the change in internal energy over time through a differential control
volume. The term 𝒒 represents the rate of heat transfer out of the control volume as a result of
heat conduction. The rate of heat transfer is determined by the following equation:
𝜕𝑇 𝜕𝑇 𝜕𝑇
𝒒 = −𝑘 ( +
+ )
𝜕𝑥 𝜕𝑦 𝜕𝑧
(7)
Equation 7 describes the flow of heat through the fluid as well as any solid structures within the
domain. [7]
9
2.1.3. Equation of State
The relationships between the variables in Equations 1 through 7 must be established in order to
fully define the flow. This is done with a series of equations that establish the thermodynamics
state of the system.
𝑝 = 𝜌𝑅𝑇
(8)
Equation 8 is the ideal gas law. It describes the relationship between the density, temperature,
and pressure of a gas based on a constant, 𝑅. [7]
2.1.4. Turbulence
Various methods have been developed for modeling turbulence in both compressible and
incompressible flow problems. One of the most common methods involves solving for the
turbulent kinetic energy (k) and the dissipation rate (ε). To reduce the complexity of the flow
model, a Reynolds Averaged Navier-Stokes (RANS) 𝑘-ε model was used. The RANS model
uses a density weighed time-average for the governing equations. This greatly reduces the
complexity of the calculation required to solve turbulent, compressible flow problems. [7]
The two equation 𝑘-ε model used to solve the flow through the combustor is as follows:
𝜌(𝐮 ∙ ∇)𝑘 = ∇ ∙ [(𝜇 +
𝜇𝑇
2
2
) ∇𝑘] + 𝜇 𝑇 [∇𝐮: (∇𝐮 + (∇𝐮)T ) − (∇ ∙ 𝐮)2 ] − (𝜌𝑘∇ ∙ 𝐮) − 𝜌𝜀
𝜎𝑘
3
3
𝜇
𝜀
2
(9)
2
𝜌(𝐮 ∙ ∇)𝜀 = ∇ ∙ [(𝜇 + 𝜎𝑇 ) ∇𝜀] + 𝐶𝑒1 𝑘 [𝜇 𝑇 (∇𝐮: (∇𝐮 + (∇𝐮)T ) − 3 (∇ ∙ 𝐮)2 ) − 3 (𝜌𝑘∇ ∙ 𝐮)] −
𝑒
𝐶𝑒2 𝜌
𝜀2
(10)
𝑘
The turbulent viscosity 𝜇 𝑇 is defined as:
𝜇 𝑇 = 𝜌𝐶𝜇
10
𝑘2
𝜀
(11)
Research into modeling of turbulent flows has resulted in widespread adoption of the turbulence
parameter values given in Table 1:
Table 1: Values used for independent turbulence parameters
𝐶𝑒1
1.44
Turbulence Model Parameters
𝐶𝜇
𝐶𝑒2
𝜎𝑘
1.92
0.09
1.0
𝜎𝑒
1.3
2.1.5. Discretization of Model Equations
The solution to the equations governing the flow through the engine cannot be directly solved in
a continuous domain. One method for solving a set of differential equations across a continuous
domain is by dividing the domain into discrete sub-domains which can be solved using
numerical methods. The solutions with these sub-domains are subsequently recombined into a
complete solution across the complete problem domain. [8]
The process of creating a set of sub-domains within the problem domain is called meshing.
Meshing is a critical step in solving a CFD problem across a domain. For a solution to be
accurate, the sub domains must be small enough to model the phenomena present. Accuracy
generally increases as the average sub-domain size decreases. [8]
The finite element method evaluates the error at the boundaries between elements for each
dependant variable and then employs a convergent iteration scheme to reduce the error. If the
mesh is sufficiently dense to model the phenomena in the domain, the solution will converge to
within an acceptable degree of error over a finite number of iterations. [8]
2.2.
2.2.1.
Modeling
Baseline Selection
To ensure that any numerical model produced realistic results, an existing engine with viable
operational data was required as a baseline. The engine selected is one research article designed
and operated by the University of Padova in Padova, Italy. [9]
The engine is a small gas turbine with a single stage compressor and a single stage turbine. The
engine is operated on a bench and the available source material does not describe operation
11
outside of a lab setting. It is designed to operate with an average turbine inlet temperature below
1000K, as this eliminates the need for high temperature alloys or ceramics in construction of the
engine. Detailed design parameters for this engine were provided, enabling a combustor with
similar operation to be modeled. [9]
However, the low temperature operation of the engine also creates difficulty when attempting to
predict NOx emissions because most publications on NOx emissions only concern flame
temperatures above 1500K. This problem was solved by the University of Padova when a
subsequent paper on the NOx emissions from this engine was published in 2009. This second
source of data allowed a relationship between turbine inlet temperature and NOx mass fraction
to be developed. [10]
2.2.2.
Combustor Modeling
The dimensions and drawings provided by the University of Padova were used to create a 3D
model of a combustion chamber and a flow region representing the interior of the engine.
A section view of the engine components provided by the University of Padova only included
some overall dimensions. To estimate the size and locations of key features of the engine
geometry, this images were imported into Microsoft Paint where a pixel count could be
conducted. By measuring the distance between dimensioned features on the figure, a pixel-permillimeter scale was determined. This scale was then used to estimate the size and location of
key geometry features in the engine. The end result of this process is shown in Figure 4.
Dimensions shown in blue were from the original image while dimensions in red were added
based on the estimated drawing scale.
12
Figure 4: Combustion Chamber Dimensional Analysis1 [9]
From these estimated dimensions, a detailed 3D model of the combustion chamber was created
in SolidWorks 2012. This is shown below in Figure 5:
Figure 5: Full 3D combustor model
1
Modified from Figure 10 of Reference [9] to color code the flow regions and show estimated dimensions
13
Initial work on simulating the full engine were unsuccessful due to the geometric complexity.
As a result, the final models used in the simulations were not meant to exactly replicate the
engine, but rather to create a simplified version with similar operating characteristics. Many
detailed features, such as the small flared tubes used to induce swirling flow, were omitted to
simplify the calculation.
This final model considered a 30 degree slice of the chamber, as the hole pattern and other
features were assumed to be axially symmetric. Many of the holes were enlarged to simplify the
meshing process and calculation. The model also includes the fuel inlet. No holes were included
on the inner wall of the chamber. The model is shown in Figure 6 and Figure 7 .
Figure 6: Combustion Chamber Simplified Model - Isometric View
14
Figure 7: Combustion Chamber Model - Side View
The air flow region encompasses the region within the interior of the engine between the
compressor outlet and the turbine inlet. The air flow region represents the overall boundaries of
the problem domain. The combustion chamber and flow region were modeled using the same
coordinate system, location, and orientation to eliminate the need for any geometry
transformations after importing them into CFD software. After importing the geometry, the
combustion chamber was already located at the proper location within the air flow region. The
air flow region is shown in Figure 8.
15
Figure 8: Engine Air Flow Region
These models were then imported into COMSOL to create a complete representation of the
domain. No additional steps were required beyond a direct import of the geometry files because
the separate entities were modeled within the same region of the coordinate system. The domain
as it appears in COMSOL is shown in Figure 9.
16
Fuel Injector
Air Flow Region
Combustion
Chamber
Figure 9: Air flow region and combustion chamber in COMSOL
The combustion chamber and the air flow region make up the extent of the domain that was
studied. However, modeling this domain in full 3D takes requires significant computer power
and time, and results in significant lag time between simulations. As a result, the flow region
and combustion chamber geometries were also modeled as 2D axisymmetric objects. This
resulted in a simpler model with greatly reduced time requirements between simulations. The
original 3D models were bisected to create 2D cross sections. These were then further
simplified and imported into COMSOL to create the axisymmetric flow domain, with the axis
of rotation at r=0. The holes were modified into slits to better represent the total cross sectional
area of flow through the combustor. The 2D axisymmetric flow domain is shown in Figure 10.
17
Fuel Injector
Air Flow Region
Combustion
Chamber
Figure 10: 2D Axisymmetric air flow region and combustion chamber in COMSOL
The operation of the engine was based on the parameters given by Benini and Giacometti. Key
parameters that were used in the simulation of the engine are shown in Table 2.
Table 2: Engine Performance: Key Parameters [9]
Property
Ambient Air Pressure
Ambient Temperature
Compression Ratio
Fuel to Air Ratio
Fuel Heating Value
Target Air Mass Flow Rate
Value
101,300 Pa
288.15 K
2.66
0.0137
42,700,000 J/kg
0.53 kg/s
Since the domain of the simulation does not include the compressor, the compressor outlet
properties were predicted as a steady state input to the interior of the engine. The density of the
air under ambient conditions was determined using the ideal gas law:
𝜌𝑎 =
𝑃𝑎
𝑅 ∗ 𝑇𝑎
18
(12)
𝐽
Where the assumed value for R was 287 𝑘𝑔∗𝐾.
The density of the ambient air is 1.225g/m3. The density of the air at the compressor outlet (the
problem domain air inlet i) was then determined according to:
𝜌𝑖 = 𝐶𝑅 ∗ 𝜌𝑎
(13)
The density at the compressor outlet was 3.2583 kg/m3. The temperature of the air at the
compressor outlet was found using an isentropic relation:
𝑇𝑖
𝜌𝑖 𝛾−1
=( )
𝑇𝑎
𝜌𝑎
(14)
Equation 14 was then solved for Ti to find that the compressor outlet temperature was 426.15 K.
The ideal gas law was used to determined the compressor outlet pressure:
𝑃𝑖 = 𝜌𝑖 𝑅𝑇𝑖
(15)
The compressor outlet pressure was found to be 398,515 Pa. These properties were used as the
conditions at the air inlet of the domain in all simulations.
The heat produced by the combustion of fuel in the air flow was determined based on the
heating value provided in Table 2 and the fuel mass flow rate determined in accordance with:
𝑚̇𝑓 = 𝐹𝐴 ∗ 𝑚̇ 𝑖
(16)
This resulted in a target mass flow rate of fuel of 0.007261 kg/s. The fuel was assumed to burn
completely within the engine, and the heat of combustion was determined in accordance with:
ℎ𝑓̇ = 𝑚̇𝑓 ∗ 𝐻𝑓
(17)
The heat generated assuming complete combustion of the fuel within the flow domain was
found to be 310,044.7 watts.
In order to simulate a reasonable amount of water being injected into the engine, a flow rate
consistent with that studied by Benini was chosen. The target mass flow rate for the atomized
water being injected into the engine was 100% of the target fuel mass flow rate of 0.007261
19
kg/s. [10] However, this mass flow rate was simulated as air entering the annular region near the
combustor exit along the inner wall of the flow domain. This was done to simplify the
numerical model and avoid the need for an accurate model of species transport. This
simplification is reasonable because the mass flow rate of water is two orders of magnitude
smaller than the overall fluid flow rate through the engine and the heat absorbing properties can
be corrected. To accomplish this, an assumption was made that the air in the simulation will
absorb an equivalent amount of heat as a proportional amount of atomized water being injected
into the engine. To determine the equivalent mass flow rate of atomized liquid water, the ratio
of the heat capacities for air and water vapor was considered. The equivalent water flow rate
was determined in accordance with:
𝐶𝑝𝑤
𝑚̇ 𝑎 = 𝑚𝑤
̇ 𝐶
(18)
𝑝𝑎
The values used for the heat capacities for water vapor and air are 1901
𝐽
𝑘𝑔∗𝐾
and 1020
𝐽
𝑘𝑔∗𝐾
,
respectively. The target mass flow rate for water entering the water injection inlet results in an
equivalent air flow rate of 0.013533 kg/s.
This value does not consider the latent heat of evaporation for the water entering the engine.
The value used for the latent heat of evaporation of water is 2,257,000 J/kg. The water will
absorb heat during evaporation according to:
ℎ𝑤̇ = 𝐻𝑤 ∗ 𝑚𝑤
̇
(19)
This results in a total heat absorption rate due to evaporation of 16,388.1 watts for the target
mass flow rate of water.
The k-ε turbulence model used in the simulations requires the turbulent intensity and turbulent
length scales to be defined as boundary conditions at the inlets. According to the COMSOL
documentation, a turbulent intensity between 5% and 10% is typical for fully turbulent flows.
The value used for all inlets is 5%, or 0.05. The hydraulic diameter at each inlet is used as the
turbulent length scale.
The boundary conditions and target flow rates in the problem domain are defined by the flow
properties at the compressor outlet. These are shown in Table 3:
20
Table 3: Problem domain inlet boundary conditions and target flow rates
Property
Air Inlet Pressure
Air Inlet Temperature
Air Inlet Turbulent Length Scale
Inlet Turbulent Intensity (All Inlets)
Fuel Inlet Temperature
Fuel Inlet Turbulent Length Scale
Water Inlet Temperature
Water Inlet Turbulent Length Scale
Value
398,515 Pa
426.15 K
0.03525 m
0.05
426.15
0.007 m
373.15 K
0.0105 m
Heat of Combustion
310,044.7 watts
Target Fuel Flow Rate
Target Water Flow Rate
Target Equivalent Air Flow Rate
Latent Heat of Evaporation for Injected Water
0.007261 kg/s
0.007261 kg/s
0.013533 kg/s
16,388.1 watts
2.2.3.
Prediction of NOx Emissions
NOx emissions increase rapidly as combustion temperature increases, particularly when the
temperature exceeds 1800 K. [6] Most available data on NOx emissions only covers
temperatures in excess of 1500 K, while the engine modeled herein in designed to operate at
about 900 K.
NOx production data according to Benini was used to determine a relationship between
combustor outlet temperature and NOx emissions as a fraction of the air compressor . [10] The
data was extracted from plots of experimental data and compiled into a table.
Table 4: NOx emissions dependence on combustor exit temperature [10]
Combustor Outlet Temperature
921 K
926 K
932 K
NOx Mass Fraction
0.0001
0.000105
0.00012
The NOx production data were then used to create an exponential curve fit for temperatures in
the range of expected operation. An exponential curve fit was chosen because it was expected to
be approximate NOx production below the temperature range reported by Benini. Even as
combustion temperature is greatly reduced, NOx production from the fuel is expected to remain
21
constant, an effect which cannot be accurately represented by a linear or polynomial fit. An
exponential fit, in contrast, provides an estimate for a minimum level of nitrous oxide
emissions. Only the 2D axisymmetric described in the following sections resulted in
temperatures significantly below the values given by Benini. The NOx produced by the engine
was approximated according to:
𝑁𝑂𝑥 = 1.950280 ∗ 10−11 ∗ 𝑒 0.01676∗𝑇
2.2.4.
(20)
2D Axisymmetric Simulation of Combustion Process
To model the combustion process, the 2D axisymmetric geometry shown in Figure 10 was
altered to include a region between the fuel injector nozzle and the combustor exit where
combustion was expected to occur. This region was used as a heat source to represent the heat
generated by the combustion of fuel. The flow domain including the combustion region is
shown in Figure 11.
22
Combustor Outlet /
Turbine Inlet
Fuel Inlet
Combustion Region
(Heat Addition)
Air Flow Inlet /
Compressor Outlet
Figure 11: 2D axisymmetric combustor model with combustion region
The combustion region and air flow domain were defined by the air material model provided by
COMSOL. The material properties were defined by curve fits to data for thermodynamic
properties. The purposed of these temperature dependant properties was to attempt to represent
real air across a wide range of temperatures and pressures.
The combustion chamber components were defined as AISI 4340 Steel, another material model
provided by COMSOL. This model accurately represented the thermodynamic properties of a
typical steel alloy as was used in construction of the research engine by Benin and Giacometti.
[9]
The domain was then meshed using the COMSOL meshing algorithm. For the problem domain
being considered, the mesh generated by COMSOL using a "normal" physics controlled setting
23
was found to provide sufficient mesh quality to solve the problem. The mesh consisted of
12,690 polygonal elements, including boundary layers to accurately predict flow behavior
along the solid surfaces within the domain. The 2D axisymmetric mesh without water injection
is shown in Figure 12:
Figure 12: 2D axisymmetric mesh, no water injection
To model the flow through the combustion chamber, the "High Mach Flow" physics modules
was applied to the problem domain. This physics module combines a compressible formulation
for fluid flow with a k-ε turbulence model and a simple heat transfer model which is sufficient
to model heat transfer across solids. The boundary condition at the compressor outlet (the inlet
of the problem domain) called for a specified inlet Mach number. However, repeated
simulations revealed that this input was used only as a starting condition and had no impact on
the problem solution after convergence. As a result, the flow rates for the various fluids entering
24
the problem domain had to be controlled by varying the boundary conditions at each of the
inlets and the outlet.
The flow field was first solved assuming no fuel flow through the engine and no heat from
combustion. The combustor exit was assumed to have a backpressure of 371,000 Pa, resulting
in a mass flow rate of 0.98254 kg/s. This was significantly higher than the 0.53 kg/s given in
Table 2. The backpressure was then evaluated at 380,000 Pa, 370,000 Pa, and 395,000 Pa,
showing decreasing mass flow rates at each value. These results were then plotted as shown in
Figure 13:
1.2
Mass Flow Rate (kg/s)
1
0.8
0.6
0.4
y = -9.10150668E-11x2 + 5.12916471E-05x - 5.51933686E+00
R² = 9.99900428E-01
0.2
0
365000
370000
375000
380000
385000
390000
395000
400000
Backpressure (Pa)
Figure 13: Air mass flow rate dependence on backpressure with quadratic curve fit
The curve fit shown in Figure 13 was generated using Microsoft Excel. It was then used to
predict the backpressure required to produce a mass flow rate through the engine of 0.53 kg/s.
The required backpressure was predicted to be 395,585 Pa. The simulation was run with this
backpressure resulting in a mass flow rate of 0.5302 kg/s. This mass flow rate was considered to
be within an acceptable margin from the baseline engine design.
Equation 16 was then used to predict the mass flow rate of fuel that would be required to
approximate the operation of the baseline engine. This was found to be 0.007261 kg/s. While
holding the air inlet and combustor outlet backpressure constant, the fuel inlet pressure was
25
varied and plotted to determine the correct fuel pressure to achieve a mass flow rate of
approximately 0.007261 kg/s of fuel. The data is plotted in Figure 14:
0.03
Mass Flow Rate
0.025
0.02
y = 0.00000633x - 2.49602687
R² = 0.99999276
0.015
0.01
0.005
0
395000
395500
396000
396500
397000
397500
398000
398500
399000
Fuel Pressure (Pa)
Figure 14: Fuel mass flow rate dependence on fuel inlet pressure with linear curve fit
This process resulted in a mass flow rate of air from the compressor of 0.52905 kg/s and a fuel
mass flow rate of 0.00695 kg/s, compared with the target values of 0.53 kg/s and 0.007261 kg/s,
respectively. These mass flow rates were considered to be within an acceptable deviation from
the target values, as they were both within 5%.
An additional model was run with the heat generated from the combustion of fuel in the engine
entering the flow domain uniformly throughout the combustion region shown in Figure 11. This
was done by adding in the "Heat Transfer in Fluids" physics module in COMSOL and coupling
the temperature variable T within the software. This physics module is formulated to account
for the effects of moving fluid on heat transfer within the problem domain. By coupling the
temperature variable between the two physics models, a solution for the flow field across the
problem domain can be found with an accurate prediction for temperature variation resulting
from the combustion process.
The boundary conditions determined without heat transfer effects were used as the initial
boundary conditions. However, the increased temperature caused the flow rates from each of the
26
inlets to change. As a result, the same process was used once again to determine the correct
pressures at each inlet, although fewer simulations were required because the boundary
conditions were still close to the correct values. The final boundary conditions and flow rates
are shown in Table 5:
Table 5: Operating characteristics for the combustor, without water injection
Variable
Air Inlet
Combustor Exit
Fuel Inlet
Pressure (Pa)
398,515
394,295
395,440
Mass Flow Rate (kg/s)
0.53075
0.53929
0.00766
The temperature distribution in the flow domain is shown in Figure 15. The average exit
temperature was found to be 708.7 K, while the peak combustion temperature was 741.06 K.
This was somewhat below the expected values, but still reasonable for the problem being
considered. Full results are reported in section 3.1.
Figure 15: 2D axisymmetric model temperature distribution, no water injection
2.2.5.
2D Axisymmetric Simulation of Water Injection
To model the injection of atomized liquid water into the combustor, an additional heat
generating region was added. This is shown in as the thin rectangular region on the inner wall of
27
the problem domain as shown in Figure 16. This region was used to subtract the heat of fusion
of water corresponding to evaporation of a percentage of the water droplets being injected.
Water Inlet
Water Evaporation
Region
Figure 16: 2D axisymmetric combustor model with combustion and water evaporation regions
The water was assumed to be saturated liquid droplets entering the engine at 373.15 K. The
evaporation process was modeled by adjusting the heat being absorbed by this region while
maintaining the inlet temperature of 373.15 K throughout the evaporation region. This resulted
in a small percentage of the water evaporating in this region. The rest of the droplets were
assumed to evaporate in the combustion region.
It should be noted that in high temperature operation, or when the water injection flow rate is
sufficiently low, all of the water droplets being injected into the engine will evaporate before
28
reaching the end of this region. In that case the heat absorbing region should be resized so that
the temperature is maintained at 373.15 K throughout.
The location for water injection was chosen in order to investigate the cooling effect of the
water on the inner wall of the chamber. Rotating machinery within the center of the engine is
used to transfer momentum from the turbine to the compressor during operation. By cooling the
inner wall of the engine, the machinery in the center can be protected from the high
temperatures created by combustion.
As with the model without water injection, the problem domain was meshed using the
COMSOL physics controlled meshing algorithm on a "normal" density setting. This resulted in
a total of 13,793 polygonal elements, including boundary layers along the walls. The mesh is
shown in Figure 17:
29
Figure 17: 2D axisymmetric mesh, with water injection
The flow field was first solved based on the inlet and outlet pressures found while modeling the
engine without water injection. The initial set of simulations did not include the heat of
combustion or the heat of evaporation of the water entering the engine. They were used to
establish the boundary conditions required for an accurate simulation of engine performance.
To determine the required boundary conditions, simulations were first run with various
combustion chamber backpressures. The combustor backpressure was varied from 380,000 Pa
to 390,000 Pa and 395,000 Pa, resulting in the range of mass flow rates from the compressor
shown in Figure 18:
30
Compressor Mass Flow Rate (kg/s)
0.9
0.8
0.7
0.6
0.5
y = -0.00001668x + 7.13147429
R² = 0.99925412
0.4
0.3
0.2
0.1
0
375000
380000
385000
390000
395000
400000
Backpressure (Pa)
Figure 18: Compressor air flow rate as a function of combustor backpressure
The relationship between backpressure and compressor flow rate was found to be nearly linear
for the range being considered. This resulted in the linear curve fit shown in Figure 18. The
relationship was used to predict the required backpressure of 395,772 Pa that was used for the
remainder of the 2D axisymmetric simulations. The next step was to find the required fuel flow
inlet pressure. Since the compressor outlet pressure and combustor backpressure were now
known, only the fuel inlet pressure was varied. Fuel inlet pressures of 395,500 Pa, 395,800 Pa,
and 395,600 Pa were simulated and resulted in the relationship to fuel mass flow rate shown in
Figure 19:
31
0.009
Fuel Mass Flow Rate (kg/s)
0.008
0.007
y = 0.0000064357x - 2.5395764286
R² = 0.9999963045
0.006
0.005
0.004
0.003
0.002
0.001
0
395450
395500
395550
395600
395650
395700
395750
395800
395850
Fuel Inlet Pressure (Pa)
Figure 19: Fuel mass flow rate as a function of fuel inlet pressure
The relationship between fuel inlet pressure and fuel mass flow rate was also found to be nearly
linear for the values considered. This resulted in the linear curve fit shown Figure 19. The
relationship was used to predict the required fuel inlet pressure of 395,736 Pa.
Now that they compressor backpressure and fuel inlet pressure were known, only the water inlet
pressure needed to be found. This was done by simulating water inlet pressures of 398,000 Pa
and 401,000 Pa, resulting in the relationship shown in Figure 20:
32
0.009
Water Flow Rate (kg/s)
0.008
0.007
0.006
0.005
y = 0.0000022200x - 0.8824100000
R² = 1.0000000000
0.004
0.003
0.002
0.001
0
397500
398000
398500
399000
399500
400000
400500
401000
401500
Water Inlet Pressure (Pa)
Figure 20: Water mass flow rate as a function of water inlet pressure
The linear curve fit shown in Figure 20 was used to determine the required water inlet pressure
of 401,700 Pa. These values were then used to simulate the flow through the combustor with the
heat of combustion and the heat of evaporation of the water droplets being considered. The
latent heat of evaporation for the water being injected into the engine was found to be 16,388.08
watts for the target water flow.
The simulation considering combustion and water injection resulted in different flow rates from
the simulation without these effects, and thus the process of determining the correct boundary
conditions once again had to be repeated. This resulted in the operating characteristics shown in
Table 6:
33
Table 6: Operating characteristics for the combustor, with water injection
Variable
Air Inlet
Combustor Exit
Fuel Inlet
Water Inlet
Equivalent Water Flow Rate
Pressure (Pa)
398,515
394,148
395,200
401,386
401,386
Mass Flow Rate (kg/s)
0.53081
0.55285
0.00726
0.01353
0.00726
The simulation was then run multiple times, and each time the amount of heat being absorbed
by the evaporating water droplets was varied until the average temperature in the water
evaporation region was found to be approximately 373.15 K. This occurred when about 6% of
the water was evaporated in the region, corresponding to a heat absorption rate of 983.3 watts.
This evaporation rate is considered valid because the heat being conducted through the wall of
the combustion chamber into this region is perfectly balanced by the latent heat of evaporation
for the water that evaporates in this region. The heat flows are therefore balanced.
The temperature distribution in the flow domain is shown in Figure 21. The average combustor
exit temperature was found to be 686.97 K, while the maximum combustion temperature was
728.7 K. Full results are reported in section 3.2.
Figure 21: 2D axisymmetric model temperature distribution, with water injection
34
2.2.6.
3D Simulation of Combustion Process
The limitations of the 2D axisymmetric model are clearly evident in the combustion
temperature results. Specifically, the values obtained are several hundred degrees Kelvin under
the experimental values. [10] To improve accuracy, a full 3D simulation was also completed.
The simulation is of a 30° slice of the problem domain. As with the axisymmetric models, a
region was added downstream of the fuel injectors to approximate the region where combustion
is expected to take place. The geometry used to model the combustion process in the baseline
engine is shown in Figure 22:
Fuel Inlet
Air Inlet /
Compressor
Combustion
Outlet
Region
Combustor Exit
/ Turbine Inlet
Figure 22: 3D combustion chamber model with combustion region
As in the 2D axisymmetric models, the fluid in the problem domain was defined as air while the
combustion chamber was AISI 4340 steel. The boundaries at either side of the 30° slice were
defined as symmetry boundaries in the software, while the outer walls and combustion chamber
surfaces were modeled with wall functions. The air inlet from the compressor stage is defined
using the boundary conditions shown in Table 3.
The combustion region is shown in Figure 22 as the 3D region just downstream of the fuel inlet.
This region was created after an initial simulation showed that the fuel flow from the injector
would swirl around this region before exiting. The shape of the region is intended to be a
general representation of where combustion is likely to occur within the engine. The fuel is
assumed to burn fully and uniformly within this region, which is defined as a heat source with a
35
total thermal power equal to 8.33% (1/12) of the value given in Table 3. Unheated fuel flow
from the injector through this region is shown as streamlines in Figure 23:
Figure 23: Fuel flow from the fuel injector through the combustion region
The domain was meshed using the built in meshing algorithm in COMSOL. First, a mesh was
generated using the "Normal" density setting. This resulted in an acceptable mesh throughout
most of the problem domain, but the software was unable to generate the default series of 5
boundary layers. The mesh was then incrementally adjusted by reducing the boundary layer
growth rate and eventually reducing the number of layers from 5 to 4. The final mesh consisted
of 380,052 volumetric elements, as shown in Figure 24:
36
Figure 24: 3D problem domain mesh, no water injection
It is important to note that like the heat of combustion, the target mass flow rates for the 3D
models are 1/12 of the total values for the engine, as the 30° slice represents 1/12 of the engine.
The target flow rates and heats rates for the 3D models are shown in Table 7:
Table 7: Target flow rates and boundary conditions, no water injection
Property
Value
Heat of Combustion
25837.1 watts
Target Compressor Air Flow Rate
Target Fuel Flow Rate
0.044 kg/s
6.051E-4 kg/s
Initially, the boundary conditions for the combustor exit backpressure and fuel inlet found
during development of the 2D axisymmetric model were used in this model. The calculated
mass flow rates were initially close to the target values, but some additional refinement to the
boundary conditions was undertaken to minimize error. The process used was similar to the
iteration technique used to solve for the required boundary conditions in the 2D axisymmetric
models. The primary limitation in refining the boundary conditions to precise mass flow rates
was simulation time.
The boundary conditions and mass flow rates for the 3D model without water injection are as
shown in Table 8:
37
Table 8: Operating characteristics for the 3D combustor model, without water injection
Variable
Air Inlet
Combustor Exit
Fuel Inlet
Pressure (Pa)
398,515
395,100
398,711
Mass Flow Rate (kg/s)
0.04325
0.0434
8.480E-4
The simulation resulted in a flow field represented by the streamlines shown in Figure 25. They
are color coded to show temperature variations as they pass through the combustion region.
Figure 25: 3D streamlines showing flow from the compressor outlet to combustor outlet, no water injection
The temperature profile shows heating of the combustion region, with hot spots forming where
flow velocities are reduced. The temperature profile is shown in Figure 26.
38
Figure 26: Temperature profile of flow through the domain, no water injection
The combustion process resulted in a peak temperature of 1748 K in localized areas, while the
average combustor exit temperature was 1036 K. This is close to the value given by Benini. [10]
39
2.2.7.
3D Simulation of Water Injection
The 3D geometry used to simulate the injection of atomized water into the engine is similar to
the 3d model without water injection, except that a heat absorbing region was added in the
space between the combustion chamber wall and the inner wall of the problem domain. The
water injection location in the 3D model corresponds to the location selected for the 2D
axisymmetric model. The geometry used to simulate water injection into the engine is shown in
Figure 27:
Water Inlet
Evaporation
Region
Figure 27: 3D combustion chamber model with heat absorbing region
The mesh generated for this geometry was similar to the model without water injection. The
meshing algorithm was set to a "normal" density setting and the number of boundary layers was
reduced from 5 to 4. The final mesh consisted of 410,101 volumetric elements, as shown in
Figure 28:
40
Figure 28: 3D problem domain mesh, with water injection
The thermal power of combustion and evaporation, and the target mass flow rates of air, fuel,
and water were all 1/12 of the values given in Table 3. The thermal powers and target mass flow
rates for this simulation are shown in Table 9:
Table 9: 3D model target flow rates and boundary conditions, with water injection
Property
Heat of Combustion
Target Compressor Air Flow Rate
Target Fuel Flow Rate
Target Water Flow Rate
Target Equivalent Air Flow Rate
Latent Heat of Evaporation for Injected Water
Value
25837.1 watts
0.044 kg/s
6.051E-4 kg/s
6.051E-4 kg/s
1.128E-3 kg/s
1365.68 watts
Initially, the pressure boundary conditions developed during completion of the 3D model
without water injection and the 2D axisymmetric model were used as the boundary conditions
for this model. The air inlet, the fuel inlet, and the exit backpressure were taken from the 3D
model without water injection while the water inlet pressure and heat absorption in the injection
region were taken from the 2D axisymmetric model results. This resulted in flow rates that were
close to the target values given in Table 9. Further refinements were undertaken to reduce error
from the target values.
41
The final boundary conditions and mass flow rates for the 3D model with water injection are as
shown in Table 10:
Table 10: Operating characteristics for the 3D combustor model, without water injection
Variable
Air Inlet
Pressure (Pa)
398,515
Mass Flow Rate (kg/s)
0.04325
Combustor Exit
395,100
0.0434
Fuel Inlet
398,711
8.480E-4
Water Inlet
395,100
0.0434
Equivalent Water Flow Rate
398,711
8.480E-4
The flow field produced by this simulation is represented as streamlines color coded to
represent temperature in Figure 29:
Figure 29: 3D streamlines showing flow from the compressor outlet and water inlet to combustor outlet
The temperature profile shows heating of the combustion region, with hot spots forming where
flow velocities are reduced. The temperature profile is shown in Figure 30:
42
Figure 30: Temperature profile of flow through the domain, with water injection
The combustion process resulted in a peak temperature of 1653 K in localized areas, while the
average combustor exit temperature was 919 K.
3.
RESULTS AND DISCUSSION
Two distinct models have been used to simulate the operation of the combustor. In the first case,
a simplified model was constructed based on a two dimensional profile which was then
revolved around the axis of the engine. This resulted in a reasonable approximation of the flow
field, while the temperature effects diverged considerably from available test data. The second
model used a three dimensional, 30° section of the engine which allowed for greater detail in
the geometry. The second model produced also produced a reasonable approximation of the
flow field, and the temperature effect closely matched available test data. This section will
report the outputs from these simulations.
3.1.
2D Axisymmetric Models
The 2D axisymmetric simulations produced flow field results consistent with available test data.
The pressure fields are shown in Figure 31:
43
Figure 31: 2D absolute pressure results without water injection (left) and with (right)
Color coding in each figure is identical. The maximum and minimum pressures are indicated at
the top of bottom of each figures color scale. The water injection caused a decrease in the
minimum pressure at the combustor exit. This is due to the increased overall mass flow. A
greater pressure differential is required to achieve the same flow rates for the fuel and
compressor air when water is added. The average pressure at the combustor exit decreased from
395,553 Pa without water injection to 394,105 Pa with water injection, a reduction of 0.37%.
The velocity magnitude of the fluid in each simulation is shown in Figure 32:
Figure 32: 2D Velocity magnitude results without water injection (left) and with (right)
44
The injection of atomized water into the engine had a small effect on the velocity of the fluid
traveling through interior of the engine. A decrease in the peak velocity near the combustor exit
was observed from 51.632 m/s without water injection to 50.176 m/s with water injection, a
reduction of 2.82%. Average exit velocity decreased from 47.11 m/s without water injection to
46.80 m/s with water injection, a reduction of 0.66%.
The temperature effects are shown in Figure 33:
Figure 33: 2D Temperature results without water injection (left) and with (right)
The addition of atomized water into engine significantly decreased the temperature of the
combustion products exiting the chamber. Peak temperature decreased from 741.06 K without
water injection to 728.7 K with water injection, a reduction of 1.7%. More importantly, average
combustor exit temperature decreased from 708.73 K without water injection to 686.97 K with
water injection, a reduction of 3.07%.
Overall performance of the engine was similar with and without water injection. The operating
characteristics in each scenario are compared in Table 11:
45
Table 11: Operating characteristics - 2D axisymmetric models
Characteristic
Air Inlet Pressure
Air Inlet Mass Flow Rate
Combustor Exit Pressure
Combustor Exit Mass Flow Rate
Average Combustor Exit
Temperature
Fuel Inlet Pressure
Fuel Mass Flow Rate
Water Inlet Pressure
Water Mass Flow Rate
Without Water
Injection
398,515 Pa
0.52962 kg/s
394,294 Pa
0.5394 kg/s
With Water
Injection
398,515 Pa
0.53081 kg/s
394,148 Pa
0.55285 kg/s
708.73 K
686.97 K
395,688 Pa
0.00882 kg/s
N/A
N/A
395,200 Pa
0.00726
401,386 Pa
0.00726 kg/s*
**This value represents the equivalent mass flow rate of water from the flow rate of fluid in the simulation
The average combustor exit temperatures were used to predict NOx emissions based on
Equation 20. The results for NOx emissions are shown in Table 12:
Table 12: 2D NOx production with and without water injection
Water Flow
Rate
0.00000 kg/s
0.00726 kg/s
Average Exit
Temperature
708.73 K
686.97 K
NOx Mass
Fraction
2.814E-6
1.954E-6
Exit Flow
Rate
0.5394 kg/s
0.5528 kg/s
NOx Production
Rate
1.518E-6 kg/s
1.080E-6 kg/s
NOx emissions are predicted by the 2D axisymmetric models to decrease by 28.84% as a result
of water injection equal to approximately 100% of the fuel flow.
In addition to the effect on NOx emissions, water injection was expected to significantly
decrease component wall temperature close to the injection site. The effect is visible in Figure
33. The wall temperature along the inner wall of the problem domain at r = -37.5 mm is plotted
relative to the z coordinate, (the water injection site is at z = 162.75 mm), in Figure 34:
46
Figure 34: 2D wall temperature without water injection (left) and with (right)
The figure shows that injecting water in this location maintains the inner wall at about 373 K for
the first 70mm as it flows into the engine. This occurs because all of the heat flux through the
combustion chamber wall is evaporating approximately 6% of the injected water in this region.
3.2.
3D Models
The 3D combustion chamber models produced flow field results consistent with the available
test data. The pressure results are shown in Figure 35:
47
Figure 35: 3D absolute pressure results without water injection (left) and with (right)
Color coding in each figure is identical. The maximum and minimum pressures are indicated at
the top of bottom of each figures color scale. The water injection caused a slight decrease in the
minimum pressure at the combustor exit. This is due to the increased overall mass flow. A
greater pressure differential is required to achieve the same flow rates for the fuel and
compressor air when water is added. The average pressure at the combustor exit decreased from
395,103 Pa without water injection to 395,097 Pa with water injection.
The velocity magnitude in each simulation is represented in Figure 36:
Figure 36: 3D Velocity magnitude results without water injection (left) and with (right)
48
The injection of atomized water into the engine had a small effect on the velocity of the fluid
traveling through interior of the engine. A decrease in the peak velocity near the combustor exit
was observed from 51.632 m/s without water injection to 50.176 m/s with water injection, a
reduction of 2.82%. Average exit velocity decreased from 47.11 m/s without water injection to
46.80 m/s with water injection, a reduction of 0.66%.
The effect of water injection on temperature within the fluid is shown in Figure 37:
Figure 37: 3D Temperature results without water injection (left) and with (right)
The addition of atomized water into engine significantly decreased the temperature of the
combustion products exiting the chamber. Peak temperature decreased from 1778.3 K without
water injection to 1653.2 K with water injection, a reduction of 7%. More importantly, average
combustor exit temperature decreased from 1037.4 K without water injection to 919.05 K with
water injection, a reduction of 11.4%.
Overall performance of the engine was similar with and without water injection, with the most
significant difference being observed for the combustor exit temperature. The operating
characteristics in each scenario are compared in Table 13:
49
Table 13: Operating characteristics - 3D models
Characteristic
Air Inlet Pressure
Air Inlet Mass Flow Rate
Combustor Exit Pressure
Combustor Exit Mass Flow Rate
Average Combustor Exit
Temperature
Fuel Inlet Pressure
Fuel Mass Flow Rate
Water Inlet Pressure
Water Mass Flow Rate
Without Water
Injection
398,515 Pa
0.04231 kg/s
372,477 Pa
0.04338 kg/s
With Water
Injection
398,515 Pa
0.0417 kg/s
395,100 Pa
0.04774 kg/s
1037.4 K
919.05 K
398,711 Pa
0.000848 kg/s
N/A
N/A
398,711 Pa
0.000840 kg/s
397,000 Pa
0.002592 kg/s*
**This value represents the equivalent mass flow rate of water from the flow rate of fluid in the simulation
The average combustor exit temperatures were used to predict NOx emissions based on
Equation 20. The results for NOx emissions are shown in Table 14:
Table 14: 3D NOx production with and without water injection
Water Flow
Rate
0.00000 kg/s
0.002592 kg/s
Average Exit
Temperature
1037.4 K
919.05 K
NOx Mass
Fraction
6.950E-4
9.560E-5
Exit Flow
Rate
0.04338 kg/s
0.04774 kg/s
NOx Production
Rate
3.015E-5 kg/s
4.564E-6 kg/s
NOx emissions are predicted by the 3D models to decrease by 84.86% as a result of water
injection at a rate of 0.002592 kg/s.
Simulation results also indicate significant cooling of the inner wall of the engine resulting from
water injection. This effect is visible in Figure 37. The wall temperatures along the edges of this
region of the problem domain are plotted in Figure 38:
50
Figure 38: 3D wall temperature without water injection (left) and with (right)
Figure 38 shows that injecting water into the inner region of the engine can maintain the inner
wall at nearly 373 K. This is because all of the heat flux through the combustion chamber wall
is evaporating approximately 1.5% of the atomized water being sprayed into the engine.
51
4.
CONCLUSIONS
This thesis has demonstrated that the injection of atomized liquid water into a gas turbine
engine reduces NOx emissions while reducing engine component surface temperatures. The
results of both the 2D axisymmetric models and the 3D models support the hypothesis that
water injection can simultaneously reduce engine component wear due to high temperatures
while simultaneously reducing pollutant emissions.
The results show that for an engine operating at low temperatures, injecting water into the
interior of the engine at a rate equivalent to the fuel flow rate reduces NOx emissions by about
30% while successfully maintaining a temperature around 373.15 K along the inner wall of the
interior of the engine. This has the effect of protecting the rotating machinery typically housed
within the center of a gas turbine engine from the heat of combustion taking place in the
combustion chamber. This can lead to significant advantages for aircraft operators because the
highest engine power output is experienced during the takeoff and climbout phases of flight and
also corresponds to the highest engine temperature and NOx emissions, making it an ideal time
to utilize water injection. [6]
The engine modeled in this thesis is a small research engine designed to operate at low (~900 K)
temperatures. [9] Typical commercial gas turbine engines operate at much higher temperatures,
where engine component wear and NOx emissions are much greater. Additionally, these
simulations only considered the heat added as a result of complete fuel combustion, and
modeled all fluids as air. A more accurate model would consider species transport of the fuel
and water being injected into the flow stream, as well as the chemical reactions taking place.
Such a model was not possible with the computer resources available.
Further research into the use of water injection for component cooling and NOx reduction
should be undertaken using commercial aircraft engine designs operating at high temperatures.
More accurate combustion models can be used to simulate the species transport and chemical
reactions taking place inside the combustion chamber if sufficient processing resources are
available.
52
5.
REFERENCES
Mattingly, J. with foreword by Ohain, H. “Elements of Propulsion: Gas Turbines and
Rockets”, AIAA Education Series, (2006)
[2] Acharya, S. " J85 ge 17a turbojet engine.jpg" Used unaltered and to develop an illustration
under Creative Commons license 3.0.
http://commons.wikimedia.org/wiki/File:J85_ge_17a_turbojet_engine.jpg
[3] “Alternative Control Techniques Document – NOx Emissions from Stationary Gas
Turbines”, U.S. Environmental Protection Agency, (1993), EPA-453/R-93-007
[4] Lefebvre, A., Ballal, D., "Gas Turbine Combustion: Alternative Fuels and Emissions"
Third Edition, CRC Press (2010)
[5] Kuo, K. "Principles of Combustion" Second Edition, John Wiley & Sons, Inc. (2005)
[6] Daggett, D., Fucke, L., Hendricks, R., Eames, D., “Water Injection on Commercial
Aircraft to Reduce Airport Nitrogen Oxides”, NASA (2010) NASA/TM-2010-213179
[7] Pletcher, R., Tannehill, J., Anderson, D., "Computational Fluid Mechanics and Heat
Transfer" Third Edition, CRC Press (2013)
[8] O.C. Zienkiewicz, CBE, FRS, "The Finite Element Method: Its Basis and Fundamentals"
Sixth Edition, Elsevier Butterworth-Heinemann (2005)
[9] Benini E, Giacometti S, Design, manufacturing and operation of a small turbojet-engine
for research purposes, University of Padova, Applied Energy Article 84 (2007) 11021116, dated 27 July, 2007
[10] Benini E, Pandolfo S, Zoppellari S, Reduction in NO emissions in a turbojet combustor by
direct water/steam injection: numerical and experimental assessment, University of
Padova, Applied Thermal Engineering 29 (2009) 3506-3510, dated 2 June, 2009
[1]
53
APPENDIX A: Determination of Boundary Conditions
Appendix A-1: 2D axisymmetric model, no water injection
Table 15: 2D model iterative convergence toward target mass flow rates, no water injection
Simulation
Number
Target
1
2
3
4
5
6
0.535
Backpressure
(Pa)
395585
395000
394294.7
394294.7
394294.7
394294.7
Simulation Values
Inlet Air Flow
Rate (kg/s)
0.53
0.50618
0.51698
0.52992
0.53061
0.52962
0.53075
Fuel Pressure (Pa)
Target Value
395479
395479
395479
395000
395688
395560
Fuel Flow Rate
(kg/s)
0.007261
0.00031
0.00327
0.0075
0.00449
0.00882
0.00766
Poly. (Simulation Values)
Mass Flow Rate (kg/s)
0.53
0.525
0.52
0.515
0.51
y = -1.84615385E-05x + 7.80928769E+00
R² = 1.00000000E+00
0.505
0.5
394200
394400
394600
394800
395000
395200
395400
395600
395800
Backpressure (Pa)
Figure 39: 2D model curve fit and predicted backpressure value to achieve the target inlet air mass flow
rate, no water injection
54
Simulation Values
Target Value
Linear (Simulation Values)
0.01
0.009
Mass Flow Rate
0.008
0.007
0.006
0.005
y = 0.0000062919x - 2.4808222804
R² = 0.9999986161
0.004
0.003
0.002
0.001
0
394900
395000
395100
395200
395300
395400
395500
395600
395700
395800
Fuel Pressure (Pa)
Figure 40: 2D model curve fit and predicted fuel line pressure value to achieve the target fuel mass flow rate,
no water injection
Appendix A-2: 2D axisymmetric model, with water injection
Table 16: 2D model iterative convergence toward target mass flow rates, with water injection
Simulation
Number
Target
1
2
3
4
5
6
7
Backpressure
(Pa)
395500
395000
394148
394148
394148
394148
394148
Air Flow
(kg/s)
0.53
0.50572
0.5147
0.52982
0.53027
0.53053
0.53071
0.53081
Fuel Pressure
(Pa)
395736
395736
395736
395400
395200
395200
395200
55
Fuel Flow Rate
(kg/s)
0.007261
0.00247
0.0055
0.01065
0.00852
0.00725
0.00726
0.00726
Water
Pressure (Pa)
401700
401700
401700
401700
401700
401500
401386
Water
Flow (kg/s)
0.013533
0.01147
0.01259
0.01448
0.01453
0.01457
0.01391
0.01353
Simulation Values
Target Value
Linear (Simulation Values)
Compressor Mass Flow Rate (kg/s)
0.535
0.53
0.525
0.52
0.515
y = -0.00001796x + 7.60890000
R² = 1.00000000
0.51
0.505
0.5
394000
394200
394400
394600
394800
395000
395200
395400
395600
Backpressure (Pa)
Figure 41: 2D model curve fit and predicted backpressure value to achieve the target inlet air mass flow
rate, with water injection
56
Simulation Values
Target Value
Linear (Simulation Values)
Fuel Mass Flow Rate (kg/s)
0.012
0.01
y = 0.0000063393x - 2.4980335714
R² = 1.0000000000
0.008
0.006
0.004
0.002
0
395100
395200
395300
395400
395500
395600
395700
395800
Fuel Inlet Pressure (Pa)
Figure 42: 2D model curve fit and predicted fuel line pressure value to achieve the target inlet air mass flow
rate, with water injection
57
Simulation Values
Target Value
Linear (Simulation Values)
0.0148
Water Flow Rate (kg/s)
0.0146
0.0144
0.0142
y = 0.0000033000x - 1.3110400000
R² = 1.0000000000
0.014
0.0138
0.0136
0.0134
401350
401400
401450
401500
401550
401600
401650
401700
401750
Water Inlet Pressure (Pa)
Figure 43: 2D model curve fit and predicted water injection pressure value to achieve the target inlet air
mass flow rate
Appendix A-3: 3D model, no water injection
Table 17: 3D model iterative convergence toward target mass flow rates, no water injection
Simulation
Number
Target
1
2
3
4
5
6
Backpressure
(Pa)
395585
395000
394294.7
394294.7
394294.7
394294.7
Inlet Air Flow
Rate (kg/s)
0.53
0.50618
0.51698
0.52992
0.53061
0.52962
0.53075
58
Fuel Pressure (Pa)
395479
395479
395479
395000
395688
395560
Fuel Flow Rate
(kg/s)
0.007261
0.00031
0.00327
0.0075
0.00449
0.00882
0.00766
0.535
Simulation Values
Target Value
Poly. (Simulation Values)
Mass Flow Rate (kg/s)
0.53
0.525
0.52
0.515
0.51
y = -1.84615385E-05x + 7.80928769E+00
R² = 1.00000000E+00
0.505
0.5
394200
394400
394600
394800
395000
395200
395400
395600
395800
Backpressure (Pa)
Figure 44: 3D model curve fit and predicted backpressure value to achieve the target inlet air mass flow
rate, no water injection
Simulation Values
Target Value
Linear (Simulation Values)
0.01
0.009
Mass Flow Rate
0.008
0.007
0.006
0.005
y = 0.0000062919x - 2.4808222804
R² = 0.9999986161
0.004
0.003
0.002
0.001
0
394900
395000
395100
395200
395300
395400
395500
395600
395700
395800
Fuel Pressure (Pa)
Figure 45: 3D model curve fit and predicted fuel line pressure value to achieve the target fuel mass flow rate,
no water injection
59
Appendix A-4: 3D model, with water injection
Table 18: 3D model iterative convergence toward target mass flow rates, with water injection
Simulation
Number
Target
1
2
3
4
5
6
7
Backpressure
(Pa)
395500
395000
394148
394148
394148
394148
394148
Air Flow
(kg/s)
0.53
0.50572
0.5147
0.52982
0.53027
0.53053
0.53071
0.53081
Simulation Values
Fuel Pressure
(Pa)
395736
395736
395736
395400
395200
395200
395200
Fuel Flow Rate
(kg/s)
0.007261
0.00247
0.0055
0.01065
0.00852
0.00725
0.00726
0.00726
Target Value
Water
Pressure (Pa)
401700
401700
401700
401700
401700
401500
401386
Water
Flow (kg/s)
0.013533
0.01147
0.01259
0.01448
0.01453
0.01457
0.01391
0.01353
Linear (Simulation Values)
Compressor Mass Flow Rate (kg/s)
0.535
0.53
0.525
0.52
0.515
y = -0.00001796x + 7.60890000
R² = 1.00000000
0.51
0.505
0.5
394000
394200
394400
394600
394800
395000
395200
395400
395600
Backpressure (Pa)
Figure 46: 3D model curve fit and predicted backpressure value to achieve the target inlet air mass flow
rate, with water injection
60
Simulation Values
Target Value
Linear (Simulation Values)
Fuel Mass Flow Rate (kg/s)
0.012
0.01
y = 0.0000063393x - 2.4980335714
R² = 1.0000000000
0.008
0.006
0.004
0.002
0
395100
395200
395300
395400
395500
395600
395700
395800
Fuel Inlet Pressure (Pa)
Figure 47: 3D model curve fit and predicted fuel line pressure value to achieve the target inlet air mass flow
rate, with water injection
61
Simulation Values
Target Value
Linear (Simulation Values)
0.0148
Water Flow Rate (kg/s)
0.0146
0.0144
0.0142
y = 0.0000033000x - 1.3110400000
R² = 1.0000000000
0.014
0.0138
0.0136
0.0134
401350
401400
401450
401500
401550
401600
401650
401700
401750
Water Inlet Pressure (Pa)
Figure 48: 3D model curve fit and predicted water injection pressure value to achieve the target inlet air
mass flow rate
62