chapter 3 introduction to cfd and problem...
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Chapter 3
INTRODUCTION TO CFD AND PROBLEM
FORMULATION
Table of Contents
S.No. Name of the Sub-Title Page No. 3.1 Introduction 48
3.2 Lagrangian Model (Dispersed Multi-Phase Flow) 51
3.2.1 Basic Conservation Equation (Dispersed Phase) 52
3.2.2 Droplet Break-up Models 56
3.3 Inlet Flow Model 58
3.3.1 Wall Impingement 59
3.4 Coupled procedure 64
3.4.1 Introduction in Solution Algorithms and Flow
Initialization
64
3.4.1.1 Common Elements 64
3.4.2 Piso Algorithm 65
3.5 Discretization Practice 68
3.5.1 General Information 68
3.5.2 Temporal Discretization 69
3.5.3 Fully Implicit Scheme 70
3.6 Error Estimation 70
3.7 Chemical Reaction and Gas Turbine Combustion
Methodology
72
3.7.1 NOx Formation 73
3.7.2 Soot Modeling 77
3.8 Problem formulation in CFD 79
3.8.1 Modeling and Meshing 79
3.8.2 Defining the Analysis 81
3.8.3 Specifying the boundaries 81
3.8.4 Thermo physical Properties 82
3.8.4.1 Reacting Flow 82
3.8.4.2 Ignition 83
3.8.4.3 Emissions 84
3.8.4.4 Additional Scalars 84
3.8.5 Lagrangian Multiphase 84
3.8.6 Boundary Conditions 85
3.8.7 Analysis Controls 86
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CHAPTER 3
INTRODUCTION TO CFD AND PROBLEM FORMULATION
3.1 Introduction
In the present investigation, the computational fluid dynamics
(CFD) has been used for the analysis in gas turbine combustion
chamber. There are many commercial CFD codes like STAR-CD,
Fluent, CFX, CFD Tutor, Numerica etc. are available in the market.
Most of these software work on the principle based on finite volume
approach. The domain has been discretized into small control
volumes by computational mesh which can statically and dynamically
distort to accommodate different types of boundaries including the
moving boundaries like piston and valves which are most complicated.
The fluid flow has to be modelled and the governing equations for the
conservation of mass, momentum, energy, and species concentration
for gas and spray droplets, together with an equation of stage, are
approximated by algebraic forms for each mesh cell. The resulting set
of equation of state, are approximated by algebraic forms for each
mesh cell. Further these set of equations are solved numerically to
obtain the gas and spray motions, the fuel evaporation, mixing and
combustion. The turbulence effects are modeled indirectly within the
Reynolds averaging framework, using high Reynolds number k
turbulence models with associated wall functions. The basic governing
equations for CFD are
49
Continuity Equation
0
kkkkk ut
(3.1)
where k - volume fraction
k - density
ku - mean phase velocity
In addition, the volume fractions must satisfy 1
d
ckk
Momentum Equation
kkkkkkk uuut
kktkkkkkk FMgp int (3.2)
where tkk and - laminar and turbulent stresses, respectively
p - pressure, assumed to be equal in both phases
kM - inter-phase momentum transfer per unit volume
kFint - internal forces
g - gravity vector
The inter-phase momentum transfer represents the sum of all
the forces the phases exert on one another and satisfies dc MM .
The internal forces represent forces within a phase. In the current
release, they are limited to particle-particle interaction forces in the
dispersed phase.
Energy Equation
Energy is solved for static enthalpy in chemico-thermal form.
50
kth
tk
kkkkkkkkkk hThuht
kktkkk
t
pk Qu
DD
: (3.3)
where kh - static enthalpy
k - thermal conductivity
kT - temperature
tk - turbulent eddy viscosity
th - turbulent thermal diffusion Prandtl number
kQ - inter-phase heat transfer
Turbulence
The present investigation consists of high turbulence swirling
flow. The flow turbulence is modeled based on the High Reynolds
number k model. Modified equations are solved for the first phase
and the turbulence of the second phase is correlated using semi-
empirical models. The additional terms account for the effect of
particles on the turbulence field.
cccccc kkt
2kcccck
tccc SGk
(3.4)
ccccccc ut
221
SCGCk cc
c
ccc
k
tccc
(3.5)
51
where ck - continuous phase turbulent kinetic energy
c - continuous phase molecular viscosity
k - turbulent Prandtl number for k equation
c - dissipation rate of ck
- turbulentPrandtl number for equation
21,CC - constants for High Reynolds number k model
cT
ccc uuuG :
And the terms 2kS and 2S represent two-phase interactions.
As the domain consists of two substances namely air and liquid fuel
droplets in the gas turbine combustion chamber it is essential to
consider the Lagrangian-Eulerian model. The detailed discussion
about the model is discussed below.
3.2 Lagrangian Model (Dispersed Multi-Phase Flow)
The flow in gas turbine combustion chamber can be analysed by
lagrangian model because it consists of a continuous phase,
comprising of liquid or gaseous and also one or more dispersed
phases which are in the form of atomized liquid fuel droplets. In
general, the motion of the dispersed phase is influenced by
continuous flow and vice versa via displacement and inter-phase
momentum, mass and heat transfer effects. The strength of the
interactions will depend on the particle size which is dispersed,
density and number density.
52
If the flow is laminar, each droplet released from injector will
follow a smooth unique trajectory, i.e. the motion is deterministic. In
present investigation, the individual elements introduced into a
turbulent carrier fluid will each have their own, random path due to
interaction with the fluctuating turbulent velocity field. The droplets
may also interact with each other (i.e. collision). When the dispersed
phase is volatile, reactive, mass transfer occurs between the phases.
This is accompanied by inter-phase heat transfer due to the inter-
phase temperature differences. Inter-phase mass transfer causes
change in the size of dispersed droplets. Thus, even if the initial size
distribution is uniform, these effects results in formation of variable-
size population. The change in size may also be due to fluid-dynamic
forces acting on the dispersed droplets which cause them to break up
into smaller droplets. Inter-droplet collisions may also produce the
opposite effect, i.e. size increase due to coalescence or agglomeration.
An additional phenomenon may occur if the dispersed droplets
strike the wall. For example, the result may be bouncing or
shattering, according to the impact conditions. The CFD code STAR-
CD provides a dispersed multi-phase flow modelling framework with
an ability to incorporate all of the above phenomena.
3.2.1 Basic Conservation Equations (Dispersed Phase)
The instantaneous fluid velocity and droplet velocity are
represented by ''u and ‘ du ’, respectively. The subscript d denotes the
droplet/dispersed phase and non-subscripted quantities are taken
which refer to the continuous phase and the droplet position vector is
53
denoted by dx . The conservation equations for droplets in the
Lagrangian framework are as presented below:
Momentum Equation
The momentum equation for a droplet of mass dm is
bampdrd
d FFFFdt
dum (3.6)
Here:
1. drF is the drag force given by
dddddr uuuuACF 21 (3.7)
where dC is the drag coefficient (which is function of the ‘droplet
Reynolds number’) and dA the droplet cross-sectional area.
2. pF is the pressure force given by
pVF dp (3.8)
where dV is the droplet volume and p the pressure gradient in
the carrier fluid.
p also includes any hydrostatic components.
3. amF is the ‘virtual mass’ force, i.e. that required to accelerate the
carrier fluid which is entrained by the droplet. The expression is as
follows:
dt
uudVCF d
damam
(3.9)
where amC - virtual mass coefficient, usually set to 0.5.
54
4. Fb is the term used to represent the general body force due to
effects of gravity and accelerations in a non-inertial coordinate frame.
In the case of rotating frame, Fb becomes
ddb urgmF 2 (3.10)
Knowledge of the droplet velocity allows its instantaneous
position vector dx which is determined by integrating the following
expression:
dd u
dtdx
(3.11)
From equation (3.7), the momentum relaxation time scale M is
identified as
dd
dd
ddd
d
dr
ddM uuC
DuuAC
mF
uum
342
(3.12)
Mass
In the presence of mass transfer at a rate mF per unit surface
area, the rate of change of droplet mass is given by
msd FA
dtdm
(3.13)
where sA is the droplet surface area. For a single-component
droplet which evaporates/condenses, mF is expressed as [3.10]
svt
vttgm pp
pppKF
,
,ln
(3.14)
where gK is the mass transfer coefficient and ,, vt pp and svp , are
the gas pressure and partial pressures of the vapour that surrounds
the droplet and also at its surface respectively. The surface vapour
55
pressure and saturation pressure is assumed to be equal at the
droplet temperature Td.
From equation [3.13 and 3.14], the mass relaxation time scale
m may be defined as
svt
vttg
dd
m
dd
ms
dm
pppp
pK
DFD
FAm
,
,ln66
(3.15)
Energy Equation
The droplet energy balance takes into consideration the
mechanisms of surface heat transfer rates dq /unit surface area and
either loss or gain that occurs due to phase change, thus:
dt
dmhqAdt
Tcdm d
fgdsddp
d , (3.16)
where sA is droplet surface area, dpc , is droplet specific heat
and fgh is latent heat of phase change. The surface heat flux, dq , is
given by the following relation:
TThq dd (3.17)
where h is the heat transfer coefficient.
From equation (3.16) and 3.17), the thermal relaxation time
scale for droplets T is determined by
NuKDc
hDc
qAmTc
m
dddpdddp
s
dddpT 66
2,,,
(3.18)
where mK is the carrier fluid’s conductivity and Nu the Nusselt
number.
56
In the present investigation the fuel which is in the form of
liquid is sprayed in the turbulent swirling air in the gas turbine
combustion chamber for vaporization and combustion. Hence, it is
important to identify the break-up model that gives the best results
from the available ones. The break-up model that has been selected is
Reitz and Diwakar model.
3.2.2 Droplet Break-up Models
Droplets become unstable under the action of the interfacial
forces which is induced by their motion relative to the continuous
phase i.e. air. There are system-specific models that exist for this
process and CFD code STAR-CD incorporates a set that are widely
used for the break-up of liquid droplets in a gaseous stream. The
different available models are:
Reitz and Diwakar
Pilch and Erdman
Hsiang and Faeth
Reitz and Diwakar model
In this model, droplet break-up due to aerodynamic forces
occurs in one of the following models:
1. ‘Bag break-up’, in this method the presence of non-uniform
pressure field sorrounding the droplet causes it to expand in the
low-pressure wake region and eventually disintegrate when
surface tension forces are overcome.
2. ‘Stripping break-up’, can be described as a process in which
liquid is sheared or stripped from the droplet surface.
57
The break-up rate can be calculated from
b
stableddd DDdt
dD
, (3.19)
where dD is the instantaneous droplet diameter. The criteria
time scales are as follows:
Bag break-up
In this method the instability is determined by a critical value of
the Weber number, We , thus:
12 bd
d Cuu
We
(3.20)
where d is surface tension coefficient and 1bC is the empirical
coefficient having a value that lies in the range 3.6 to 8.4 and in CFD
code STAR-CD the value of 1bC =6 is taken as the default setting. The
stable droplet size can be defined as the one that satisfies the equality
in the above equation.
The associated characteristic time is
21
23
21
2
4 d
ddbb
DC
(3.21)
in which 2bC .
Stripping Break-up
The criterion for the onset of this regime is
1Re sd
CWe (3.22)
where dRe is the droplet Reynolds number and 1sC is a
coefficient which has a value of 0.5.
58
The characteristic time scale for this regime is
d
ddsb uu
DC
21
2
2 (3.23)
Here, the empirical coefficient 2sC for the star-CD with a default value
of 20. The effect of inter droplet collisions has not been taken into
consideration.
In the present work atomized liquid fuel droplets is injected into
turbulent swirling flow of air in gas turbine combustion chamber. It is
important to select a suitable model suitable for our application.
3.3 Inlet Fuel Flow Model
There are different methods of fuel injection i.e. point injection
and nozzle injection as both the process being includes the flow either
in the point injector or the in the nozzle hole and also their
atomization. This gives rise to a fuel droplet spray represented by a
set of computational parcels, each containing a number of droplets
with the same properties. These properties are calculated on the
basis of various atomization models. The histories of the
computational parcels are obtained from the Lagrangian conservation
equations of mass, momentum and energy.
Point Injector Model
The fuel injection velocity plays a prominent role when it
sprayed in gas turbine combustor. It plays a pivotal role in
atomization, break-up processes and as a consequence influences the
fuel penetration, vaporization, inter-droplet interactions, droplet-wall
59
interactions. Hence each injection point can be specified by its
location and can be assigned with injection velocities both in axial
direction (along the axis of combustor) and radial direction (rotational
velocity), droplet particle size function, fuel injection angle, number of
droplet parcels, mass of fuel that has to be injected can be defined.
Nozzle flow model
The injection velocity, chu , i.e. the velocity of the liquid fuel as it
comes out of the nozzle and enters the combustion chamber, is
considered as one of the most important parameter in the spray
calculation. It highly influences different processes like atomization,
break-up, spray penetration inside the combustor, inter-phase
transfer and also the interactions between droplet-droplet and droplet-
wall.
After the fuel is sprayed in the combustor, it is desired that
these droplets vaporize and later the combustion occurs. But in many
cases it is found that some of the droplets sprayed collide into
adjacent solid wall. Hence it is necessary to select a suitable model for
walls impingement models. In the present case Bai’s spray
impingement model is found to be most suitable for our application
and it has been selected.
3.3.1 Walls Impingement
The collision of droplets on rigid solid surfaces produces a wide
range of consequences which is based on the size, velocity and
material of the impacting elements and the nature of the surface. For
instance, droplets may stick, rebound or shatter and the liquid that
60
deposits on the surface may combine into a liquid film or retain its
droplet form.
CFD code STAR-CD offers the following optional treatments of
wall impact:
Perfect rebound
Adherence, either in the form of a liquid film or as spherical
droplets. Droplet that gets adhered to the walls will exchange
heat with them and it also continues to interact with the air or
any carrier fluid as far as heat and mass transfer is concerned.
Instantaneous evaporation
Bai’s spray impingement model
This spray impingement model has been formulated within the
structure of the Lagrangian approach, on mass, momentum and
energy conservation constraints. As it reflects the stochastic nature of
the impingement process, a random procedure has been formulated to
determine some of the droplet post-impingement quantities. This
allows secondary droplets generated from a primary droplet splash to
have a distribution of sizes and velocities.
The CFD code STAR-CD implements with following assumptions
either dry or wet wall impingement
elastic rebounding from walls when temperature is higher than
the critical/boiling temperature
There are varieties of impingement regimes which are governed
by a number of parameters such as incident droplet velocity relative to
the wall, incidence angle, droplet size and properties as well as wall
61
temperature, wall surface roughness, liquid film thickness, near-wall
gas conditions.
Some of the physical quantities involved are combined into
dimensionless parameters, e.g.
This is a variant of the droplet Weber number dWe which
uses droplet density and also the normal-to-the-wall
components of relative velocities between all and droplets
the Laplace number, aL , defined as
2d
ddd DLa
(3.24)
Regime transition criteria
The model characterizes the following regimes:
Stick – the impinging droplet sticks to the wall in nearly
spherical form
Spread – the droplet spreads out along the wall surface
creating a wall film or merging with an existing film on a
wetted wall
Rebound – the impinging droplet bounces off the wall after
the impact
Splash – the impinging droplet breaks up into smaller
droplets, some of which are reflected from the wall.
For the sake of completeness, the transition criteria for both dry
and wetted walls are given below:
62
1. Dry wall
(a) Adhesion (combine with stick and spread
regimes):
ad WeWe
Where aWe is:
18.0 aa ALWe
(b) Splash:
ad WeWe
Table 3.1: Dependence of Coefficient A on the wall surface
roughness
mrs 0.05 0.14 0.84 3.1 12.0
A 5264 4534 2634 2056 1322
After discussing the different wall impingement models it is
necessary to define probability density function that define the droplet
size distributions. This is very important as the sizing of droplet plays
an important role in vaporization and combustion processes. The
Rosin -Rammler method has been adopted in our investigation.
Probability Density Functions for initial droplet size
distributions
It has been discussed different atomization models that have
been employed to determine the initial droplet size in the vicinity of
the injector nozzle. But suitability of some of models may not be
possible to all types of injectors and nozzles as some of them use their
63
own data when it is available. Also large amount of data for various
types of nozzles has been collected mostly by the nozzle
manufacturers and the researchers and they use them. The
applications and the collected data can be correspondingly correlated
into a number of size distribution functions. The users give topmost
priority to utilize these functions in order to obtain accurate
description of initial droplet sizes for their applications.
But some of the mathematical and experimental size
distributions are available
1. Rosin Rammler Method:
q
XDQ exp1 (3.25)
where Q is the fraction of total volume occupied by the droplets
with their diameter less than D. Also, X and q are the empirical
constants that will be specified by the user.
2. Normal Distribution Method:
2
221exp
21)( DD
SSDf
nn
(3.26)
where Sn is described as the deviation of diameter D with
respect to the mean diameter D that can be referred to as standard
deviation. The values of Sn and D will be specified by the user.
This equation when integrated gives cumulative probability
density function and later this is used to determine the size
distribution of droplets that has been released by taking random
samples, each one with a probability of �i (0<�i<1).
64
The domain which has been modeled, meshed it is necessary to
define the boundary conditions and the method the solution method
that have to be followed so as to obtain a better solution.
3.4 Coupled Procedure
3.4.1 Introduction to Solution Algorithms and Flow
Initialization
The CFD code STAR-CD currently incorporates two different
implicit algorithms, namely:
1. A variant of the well-known SIMPLE (Semi-Implicit Method for
Pressure-Linked Equations) method which is generally used for
the cases involving steady state solutions.
2. The more recent and efficient PISO (Pressure Implicit with
Splitting of Operators) method which is generally used for the
cases involving transient solutions.
3.4.1.1 Common elements
The algorithms of both SIMPLE and PISO have common features
and the details are discussed below
1. Both of them use different forms of predictor-corrector method
as operator splitting has been enabled to decouple the flow
equations temporarily from each other so that they can be
solved sequentially.
2. Continuity has been implemented with the help of an equation
set for pressure that has been derived by combining different
equations of finite volume momentum and mass conservation
equations.
65
3. The solution follows the order that involves a predictor stage,
which produces a temporary velocity field that has been derived
from the momentum equation and a provisional pressure
distribution. These temporary fields are later refined in the
corrector stage by demanding simultaneous satisfaction, to a
pre-determined level of approximation, for both momentum and
continuity balances.
4. For the iterative calculations to the steady state, the sequence of
steps discussed above are executed repeatedly with the
incorporation of updating the field-dependent coefficients which
are obtained from coupling and nonlinearity till the final
solution has been obtained. As a measure for the stabilization of
this process under-relaxation has been employed.
5. Within the sequence, of the operator-split equation sets that has
been engaged at any stage, only one of the field variables, i.e.
the vector set of unknowns has been split into a sequence of
scalar sets.
6. The algebraic equation sets have been solved by iterative
methods though it is not an essential practice.
3.4.2 The PISO algorithm
The above procedure is repeated in the PISO corrector stages at
the end for which the dispersed elements are assigned their final
positions, velocities, etc. for the interval t . The next time step is then
commenced with PISO, a predictor only procedure may also be used to
66
bypass the corrector stages completely. This will significantly speed up
the computation but at the expense of solution accuracy.
Working equations
For the presentation, the equations of finite volume momentum
has been extracted from the general transport equation which are in
the form
nN
nNP
oPi
oP
nmi
nPiP ppDsuBuHuA 1,,, (3.27)
where
mimmi uAuH ,, (3.28)
Fig: 3.1 Arrangement of variables and notation for PISO
implementation on Cartesian mesh
Solution sequence
The program starts with from initial value o of the variable
fields, PISO progresses along with a time increment t in the
subsequent sequence of steps.
P uiP
uj
67
1. Predictor stage equation [3.27] has been assembled and solved
in the following operator-split form for the momentary nodal
velocity field 1iu
oN
oNP
oPi
oP
nmi
nPiP ppDsuBuHuA 1,,, (3. 29)
where oP corrector to the pressure field at the
commencement of the time step. After getting the solution of
this equation which has been obtained by an iterative method,
the momentary face velocities 1ju has been calculated, with the
niu and nP replaced by 1
ju and oP , respectively.
2. First corrector stage. The operative nodal momentum equation
is now taken to be
111,
1,
2, NNP
oPi
oPmiPiP ppDsuBuHuA (3.30)
and the face momentum equations are similarly approximated
by replacing nju and P in the equation 3.29 by 2
ju and 1P ,
respectively. Correspondingly, the pressure equation in its
approximated form reads.
1
1,
2 spAuPAm
mmmipp (3.31)
3. Additional Corrector Stages. Further corrector stages are
calculated in the similar manner as the first one, using the
generalized equations.
qN
qNP
oPi
oP
qmi
qPiP ppDsuBuHuA 1,,
1, (3.32)
1, spAuPA
m
qmmmi
qpp (3.33)
68
where ,.....2,1q is the corrector level, PA are held
constant. The solutions from the consecutive stages represent
progressively better approximations to the solution of the initial
equations, i.e. 1qju and 1qP tend towards n
ju and nP with
increasing q .
After completion of the desired number of correctors, judged in
keeping with practices mentioned above, the solution obtained are
taken as inputs to the starting field for subsequent time step and also
the sequence is repeated from stage 1. Calculation of scalar fields
such as the turbulence parameters and temperature will be done if
needed but additional steps are to be performed in the final flow
corrector. On the other hand if the scalars are strongly coupled to the
flow field like effects of compressibility or combustion, they are
embedded within the main predictor/corrector sequence.
3.5 Discretisation Practice
3.5.1 General Formation
The differential equations governing the conservation of mass,
momentum, energy, etc. within the fluid are discretized by the finite
volume method. Thus, they are integrated over the individual
computational cells, then approximated in terms of the cell-centered
nodal values of the discretized forms preserve the conservation
properties of the parent differential equations.
For the purpose of the finite volume discretization, the
conservation equations:
69
sgradudivt
r
(3.34)
where
cr uuu is the relative velocity between the fluid
u the local coordinate velocity
cu stands for any of the dependent variables (i.e. 1,,, mkeui etc.) and
s, are the associated ‘diffusion’ and ‘source’ coefficients, which can
be deduced from the parent equations.
An exact form of equation (3.34) valid for an arbitrary time-
varying volume V bounded by a moving closed surface S can be
written as:
dVsSdgradudvt VS
rV
(3.35)
where
S is the surface vector and ru
is now the relative velocity
between the fluid
u and the surface
cuS . If V and S are,
respectively, taken to be the volume pV and discrete faces fj NjS ,1
of a computational cell equation [3.34] becomes
dVsSdgradudvt Vj S
rVp
(3.36)
3.5.2 Temporal Discretisation
The finite volume equation has been applied over arbitrary time
increment t spanning the ‘old’ and ‘new’ time levels. The temporal
discretisation has been existing for two methods. The first one is the
70
default method which is a fully-implicit scheme. The second method is
a second order Crank-Nicholson scheme that may also be used.
3.5.3 Fully implicit scheme
In this method, the fluxes prevailing over the specified time
interval are calculated from the new time-level values of the variables.
In principle, the fully-implicit formulation allows any magnitude
of your time step to be used, however for different applications other
considerations impose limits i.e. for transient problems, t should be
enough to limit the temporal approximation errors to acceptable
levels.
3.6 Error Estimation
The error estimation offers a summary of the approximate
magnitude and distribution of the discretisation error within the
solution domain. The errors that have been taken into consideration
in the estimate are that caused by the mesh spacing, irregularity and
non-orthogonality, the selection of the convection differencing scheme
and also includes the time-step size for transient calculations.
The method of estimation of residual error that has been
implemented in the CFD code STAR-CD is based on the local
imbalance between the face interpolation and volume integration,
which produces a cell residual. Then the residual obtained is
normalized in a proper way to produce the estimate of absolute
magnitude of the error, with similar physical dimensions in response
to the variable in question.
71
For the Navier-Stokes equations, the estimation has been done
in velocity field errors also with the error in pressure distribution
incorporated into the same estimate. Even though the residual error
estimate provides the error in vector form, it is generally more adapted
to look at its magnitude.
In turbulence distribution, discretization error magnitude can be
obtained from the estimated error within the turbulent kinetic energy
and velocity. If the near-wall regions have been assumed as near wall
functions, there is a possibility of presence of some interference with
the error estimation method which depends on the y values and
quality of local mesh. If the residual error that has been estimated for
velocity produces an unrealistically high error near the wall, a special
inbuilt procedure has been incorporated to correct it. Depending on
the characteristics of the flow, the error estimate in temperature,
turbulence dissipation and different scalars can also be enclosed.
After the modeling and assigning the boundary conditions to
different boundaries, there are reactions that occur during the
combustion process in the domain. It is essential to incorporate the
chemical reactions along with the conditions at which these reactions
occur. Also, when the reaction occurs is in the form of combustion
resulting formation of products of combustion and thereby releasing of
enormous amount of energy.
72
3.7 Simulation methodology of chemical reaction in the
gas turbine combustor
The CFD code STAR-CD has been for handling chemically
reacting flows as it provides for the solution of conservation equations
for chemical species and energy. This allows possible connection with
the flow field via density changes brought by the temperature and
variations in mass fraction.
Most of the physical complications are the issues associated
mainly with chemical kinetics, turbulence and their interactions. The
mechanisms are related to the chemical kinetics and at molecular
level the rates of reaction of chemical species.
For gas turbine problems, the specialized modeling capabilities
are necessary and available in CFD code STAR-CD. It is required to
perform combustion calculations in specific to gas turbines, the
facilities that are available have been taken into account with the
following features of engine process:
Fluid dynamics, including turbulence
Heat transfer
Multi-component gaseous mixing
Momentum, heat and mass exchange with droplets
Combustion
Complex geometry
One of the most common assumptions in flame-area models is
that the flame has extremely thin surface which propagates in space
73
as it is affected by different physical processes, such as stretching,
curvature. These processes have been modeled through a transport
equation for the flame density or a variable which is related and that
describes the distribution of the flame surface.
3.7.1 NOx Formation
One of the main air pollutants are oxides of Nitrogen which are
prominently produced by combustion devices. It is seen in most of
devices that the concentration of NOx is low and hence it has very
small influence on the flow field. Further the time scale for NOx
reaction is very large than the time scales of turbulent mixing process
and the hydrocarbons combustion as it regulates the heat-releasing
reactions. Hence from the main reacting flow field predictions the
computations of NOx can be dissociated.
Three different mechanisms have identified for the formation of
nitric oxide during the combustion of hydrocarbons, namely:
1. Thermal NOx
It is strongly temperature dependent. It is produced when
atmospheric nitrogen with oxygen reacts together at higher
temperatures.
2. Prompt NOx
The formation of prompt NOx details is still uncertain and the
general perception is that the reactions involving atmospheric nitrogen
and hydrocarbon radicals lead to its formation. In certain cases
involving combustion favourable conditions like low-temperature, fuel-
74
rich condition and short residence time, prompt NOx can be produced
in significant quantities.
3. Fuel NOx
This is produced by the reaction of the nitrogenous elements
that are present in the fossil fuels both liquid or solid with oxygen.
The fuel nitrogen is a vital source of nitrogen oxide emissions for
residual oil and coal, which generally contain 0.3 – 2.0% nitrogen by
weight.
The presence of nitrogenous components in the fuel either liquid
or solid forms reacts with oxygen resulting in formation of fuel NOx.
The important source in formation of nitric oxide for coal and fuel oil
is the presence of fuel nitrogen which generally contains 0.3 – 2.0%
nitrogen by weight.
Thermal NOx
The presence of high temperature is a favourable condition for
the reaction between atmospheric nitrogen and takes place as a result
formation of thermal NOx occurs. The main reactions for formation of
thermal nitric oxide have been proposed by the following three
extended Zeldovich mechanisms:
NNOONK
K
1
12 (3.37)
ONOONK
K
2
22 (3.38)
HNOOHNK
K
3
3
(3.39)
75
The rate constants for the above reactions have already been
measured in several experimental studies, the data which has been
obtained from these studies is evaluated by Baulch. The expressions
calculate the rate coefficients for above reactions are given below:
TXK 38370exp108.1 11
1 m3 (kg mol)-1 s-1 (3.40)
T
XK 425exp108.3 111 m3 (kg mol)-1 s-1 (3.41)
TTXK 4680exp108.1 7
2 m3 (kg mol)-1 s-1 (3.42)
T
TXK 20820exp108.1 62 m3 (kg mol)-1 s-1 (3.43)
TXK 450exp101.7 10
3 m3 (kg mol)-1 s-1 (3.44)
T
XK 24560exp107.1 113 m3 (kg mol)-1 s-1 (3.45)
where 1K , 2K and 3K are the forward and 1K , 2K and 3K are the
backward rate constants for reactions, respectively.
The breakage of strong N2 bond takes place by thermal fixation
of nitrogen which occurs at higher temperatures with the rate of
formation of NOx has been substantial only. This effect has been
represented by the high activation energy of reaction, which is a key
reaction as it is the rate-limiting step of the Zeldovich mechanism.
The energy required for the activation of oxidation of nitrogen is quite
small thereby establishing the quasi-steady state. On the basis of this
assumption, the instantaneous of NOx is:
76
OH
OH
O
O
NO
NO
MYK
MY
KMYK
R
321
2
2
2/1
NO
NO
H
H
NO
NO
O
O
OH
OH
O
O
NO
NO
O
N
O
O
MY
MYK
MY
MYK
MYK
MY
K
MYK
MY
MYK 32
32
1
1
2
22
2
22
kgmol m-3 s-1 (3.46)
If the mass fraction of radicals O, OH and H are calculated by
the combustion model, they are used in the NOx model. Otherwise,
certain assumptions have to be made to obtain their values. For
example, according to Westenberg, the equilibrium O atom mass
fraction can be obtained from:
21
2OKO p kgmol m-3 (3.47)
and
OO
MOY (3.48)
where
TTXK p
31090exp1097.32
1
5
(kgmol)1/2 m3/2 (3.49)
In the mass fraction of O atoms is obtained from equation (3.47)
and those of OH and H are initially set to zero.
Prompt NOx
The formation of Prompt NOx occurs from reaction of molecular
nitrogen and hydrocarbon fragments. The lifetime is for a period of
only several microseconds and weak temperature dependent. The
77
contribution to NOx emission from this mechanism is higher in the
systems which produce very fuel-rich flame (e.g. staged combustion
system). In utility furnaces, the contribution of prompt NOx to the
total NOx is extremely small in comparison with fuel NOx. The role of
prompt NOx but is predicted to rise with NOx emission levels being
reduced continuously with the use of latest combustion technologies.
The production rate of prompt NOx is given by De Soete as,
NOptPtNO MkR , kg m-3s-1 (3.50)
Where ptk is a rate constant and is defined as:
RTEFuelNOkk ab
rpt exp22 kgmol m-3s-1 (3.51)
2O , 2N and Fuel are species concentrations in kg mol/m3.
For CH4 air flames, the activation energy aE is about 303.5X106J/kg
mol. The universal gas constant R and temperature T and are
represented in SI units. The oxygen reaction order, b is mainly
dependant on the experimental conditions. De Soete had described
that b varies with the oxygen mole fraction 2OX as follows,
0.1b 0041.02OX
2
ln9.095.3 OXb 0111.00041.02 OX
2
ln95.3 OXb 03.00111.02 OX
0.0b 03.02OX
3.7.2 Soot Modeling
The formation and emission of carbonaceous particles is a
process that is often observed during the combustion of
78
hydrocarbons. These particulates, called soot, are identified in flames
and fires as yellow luminescence.
It is required to have a detailed knowledge of the different
mechanisms that leads to formation of soot. In general it is widely
accepted that the process of soot formation is highly complex in
nature which comprises of
fuel pyrolysis and oxidation reactions
formation of polycyclic and aromatic hydrocarbons
the inception of first particles
the growth of soot particles due to reaction with gas phase
species
the coagulation of particles
the oxidation of soot particles and intermediates
Mathematical model
In this model, an additional transport equation is solved for the
mass fraction of soot. The flamelet based approach has been used for
modelling of the soot. The source terms for the soot volume fraction
have been taken from a flamelet library employing a presumed
probability density function and integrating these terms over mixture
fraction space. To save CPU time and computer storage, the flamelet
library of source has been constructed by employing a multi-
parameter fitting procedure that has resulted in simple algebraic
equations and with a proper set of parameters.
The transport equation for soot mass fraction is given by
79
vsj
s
st
t
jsj
js x
Yx
Yux
Yt
,Pr
(3.52)
Where SY is the soot mass fraction. The Prandtl number for soot is
assumed to be 1.4 and the soot density s =1860 kg/m3.
3.8. Problem formulation in CFD
The sequence of steps that has been followed to set up the
model in CFD has been discussed in detail in the following sections.
3.8.1. Modeling and Meshing
The first step is designing a fluid domain model for the analysis
using CFD [128]. The geometric model of a combustor is shown in Fig:
3.1. This is modeled using Star Design software.
Fig: 3.2 Geometrical Model of Combustor
The next step is the process of mesh generation which has been
a challenging task to users of CFD in combustor modeling. This has
been difficult and time consuming, because the mesh generation
technique requires considerable expertise and effort to address
meshing of the combustion chamber. It is to be noted that the fuel is
R θ z
Atomiser
Helicoidal Swirler
Secondary Air Dilution Air
0.8 D
2 D
4 D
D
Primary Air
80
sprayed at the axial entrance of the combustor located near the axis.
In this region the mesh size chosen is very fine so as to capture all the
information pertaining to fuel droplets i.e location of fuel droplets,
vaporization and combustion of the charge commences at the
entrance and slowly dies down as it moves to the exit of combustor
thereby increasing the temperature. Hence, the mesh had to be
adjusted to accommodate the different sizes of the cells being fine at
the entrance and the axis of the combustor and coarse at exit and
periphery of the combustor as depicted in Fig: 3.2.
Fig: 3.3 Meshed model of Combustor
So two separate mesh domains are generated and the
connectivity between these two domains has been established while
ensuring that the cell does not become excessively distorted. The
whole solution domain is broken up into many face-connected
polyhedral control volumes or cells. The structure that is formed by
combining all the cells together is referred to as mesh. The domain
has been assumed as axis-symmetric model comprising of 7 sectors,
each with an angle of 51.42o and the total amounting for 360owith
seven sets of radial air inlets. One such sector with an angle of 51.42o,
Secondary Air Dilution Air
Primary Air Inlet
Outlet/exit
81
has been modeled and meshed with the 162137 number of cells for
the present analysis.
3.8.2 Defining the Analysis
The turbulent swirling air starts to flow in the combustor and
after it gets stabilized the atomized liquid fuel droplets is sprayed into
the air stream of turbulent swirling air. The combustion commences
after ignition and attains a steady state in due course of time for a
constant load conditions. Hence the analysis is carried out for steady
state condition. There are two phases involved i.e. the stream of air in
gaseous phase and atomized fuel droplets in the liquid phase and
hence the type of flow considered is multiphase. The process of
combustion of fuel occurs in the presence of air involving chemical
reaction and hence the flow has been considered as reacting flow with
the involvement of chemical reactions.
3.8.3 Specifying the boundaries
The domain boundaries are specified in terms of different regions
which comprises of air inlets one located at the axial entrance called
primary inlet, two in radial direction called secondary and dilution
inlets. The other boundaries are outlet at exit of combustor located
opposite to the axial air inlet, two sides along the axis of combustor
are considered as symmetry (connects the other sectors on either side)
and the rest of them as walls. Also a portion of axial air inlet is
considered as a wall (between axis and hub of swirler).
82
3.8.4. Thermo physical Properties
There is variation in density and specific heat of inlet air with
temperature and it has been taken as polynomial function of
temperature and pressure with the molecular weight of air specified as
28.96 Kg/mol. The mass flow rate of air is high and the turbulence
model is considered as k-ε model with high Reynolds number. There is
a rapid rise in temperature that occurs during the combustion of fuel
and the thermal model with an option of static enthalpy and chemico-
thermal has been taken. The initial values of air are assigned in the
initialization process.
3.8.4.1 Reacting Flow
The chemical reaction as it involves the oxidation process of
fuel. The type of reaction selected is a non-premixed as the atomized
fuel droplets are sprayed in the combustor. The type of reaction model
selected for the combustion involving the liquid fuel spray is
Magnussen eddy break-up model. This model is one of the earliest
models used for turbulent chemical reaction which has been
successful over the years and extensively used even till today. This
model has been developed specifically for the specialized application in
combustion. The assumptions made in this model for analysis are
discussed below:
This is a single step irreversible reaction involving fuel, oxidizer
and the products of combustion along with the possible
background of inert species.
Ex: Fuel + Oxidizer Carbon Dioxide + Water Vapour
83
The reaction timescale is so small and hence the mechanism of
rate of combustion occurs by turbulent micromixing.
The reaction involving n-Dodecane (C12H26) as fuel and oxidizer
as oxygen results in formation of product like carbon dioxide and
water vapour is specified in the reaction system. The corresponding
constants involved for this reaction are taken as default values which
are specified by the Magnussen model used for combustion.
3.8.4.2 Ignition
The combustion takes place after ignition occurs. So, some of
the cells in the domain are specified as ignition cells generally located
above the fuel injection which act as hot spots for the initiation of
ignition. These cells are active for a very limited period and they can
be controlled by specifying its activation and deactivation in terms of
iterations. The ignition cells in the domain are sky blue in colour as
shown in Fig: 3.3.
Fig: 3.4 Ignition cells in the combustor
Secondary Air Inlet
Primary Air Inlet
Ignition Cells
84
3.8.4.3 Emissions
With the initiation of ignition the combustion commences and
the formation of products of combustion and emissions takes place.
The emissions are mainly Nitric Oxide, Soot and Carbon Monoxide.
The emissions of thermal NO, Prompt NO options are activated so that
they collect the information about the formation of NO. Similarly Soot
model option is activated to track the formation of soot during the
process of combustion.
3.8.4.4 Additional Scalars
The inputs about the air, fuel have been specified earlier and
the software generates number of additional scalars that are
generated automatically with some of their influences are in active
mode (direct control) and rest of them in passive mode (indirect
control). The details of the scalars with their influences in active mode
are Oxygen (O2) and Nitrogen (N2) present in air, n-Dodecane fuel for
combustion, Carbon dioxide (CO2) and Water Vapour (H2O) are the
main products of combustion.
3.8.5. Lagrangian Multiphase
The atomized fuel droplets sprayed in the combustor can be
tracked by two methods. In Eulerian approach mean of all droplets is
taken into consideration for the analysis and the solution obtained is
faster. But in Lagrangian approach each particle is given a separate
identity and the path traced by each droplet is taken into
consideration. Hence the memory and computational time
requirements is quite large. Also this method is used for the cases
where detailed investigation is necessary and for research works. In
85
the present investigation Lagrangian approach is used. The memory in
terms of gigabytes is allocated to the droplet trajectory file. Similarly
the total number of droplet parcels taken for the simulation is
specified where the injection of these parcels is defined explicitly.
These droplet physical models are subjected to mass transfer,
momentum transfer with turbulent dispersion. The droplet breakup is
governed by Reitz-Diwakar method and the droplet wall interactions
are governed by Bai’s impingement model. The fuel n-Dodecane in a
liquid phase is selected as a single component sprayed in the
turbulent swirling flow of air where the evaporation of these droplets
takes place as defined in the scalar.
The Rosin-Rammler method defines the droplet size sprayed in
the combustor with the input values assigned to parameter X and
parameter q. Further the fuel injection velocity is assigned the values
according to the requirements. The swirl component is assigned to the
fuel wherever necessary and the fuel injection temperature is assigned
the value of 350 K.
A number of fuel injection points are selected such that the
product of these points and the number of droplet parcels for injection
point is equal to total number of droplet parcels. The locations and the
angle of fuel injection points are specified accordingly.
3.8.6. Boundary Conditions
Earlier we had specified the boundaries in terms of regions and
in this section we assign some conditions and also values to them. All
the inlet boundaries are specified as velocity boundaries as the
compressed air enters the combustor with the same velocities through
86
all inlets. The air passes through the swirler and enters the combustor
in axial direction hence the swirl component is added to the velocity
assigned at the axial air inlet, whereas the radial air inlets are
assigned the same velocities without any swirl component. The values
of turbulent intensity, density of air, temperature of air have been
assigned. All the boundaries which are considered as wall are
assigned to be adiabatic in nature. The outlet is treated as standard
outlet where the products of combustion are vented out to
atmosphere. The values of Nitrogen and Oxygen that are present in air
are assigned at each inlet.
3.8.7. Analysis Controls
The method of solution is assigned as steady state with the
solution algorithm followed is SIMPLE (Semi-Implicit Method for
Pressure-Linked Equations). The solver type is specified as scalar
whereas the solution method is defined as CG (Conjugate Gradient).
The solver parameters like temperature and density have been
assigned a suitable relaxation factor and residual tolerance. The
differencing schemes for flow variables U- Momentum, V-Momentum,
W-Momentum, Turbulent kinetic energy, Turbulent dissipation,
Temperature is taken as Upwind Scheme and for Density it is taken
as Central Differencing Scheme with a suitable blending factor. There
are some other controls for various parameters predefined in the form
of switches which are switched on and some of the real constants are
assigned some values. With above sequence of steps followed, the
domain with all the inputs completely specified is employed for the
analysis.