computational work for compressible flow

Remark: foils with „black background“
could be skipped, they are aimed to the
more advanced courses
Rudolf Žitný, Ústav procesní a
zpracovatelské techniky ČVUT FS 2010
Combustion,
multiphase flows
reduced…
Computer Fluid Dynamics E181107
CFD8r
2181106

COMBUSTION
CFD8r
Homogeneous
reaction in gases
Premixed (only one
inlet stream of mixed
fuel and oxidiser)
Non premixed
(separate fuel and
oxidiser inlets)
Laminar
flame
Turbulent
flame
Laminar
flame
Turbulent
flame
Use mixture
fraction method
(PDF)
Use EBU (Eddy
Breakup models)
Liquid fuels (spray
combustion)
Combustion of
particles (coal)
Lagrangian method-trajectories of a
representative set of droplets/particles
in a continuous media
Lagrangian method-trajectories of a
representative set of droplets/particles
in a continuous media

COMBUSTION aims
CFD8r
Primary purpose of CFD analysis is to evaluate
Temperature field (therefore thermal power, heat fluxes through wall…)
Composition of flue gas (environmental requirements, efficiency of burning)
To do this it is necessary to calculate
Velocities and turbulent characteristics (mixing intensity) – NS equations
Transport of individual components (mass balances of species)
Chemical reactions (reaction rates)
Energy balances (with special emphasis to radiation energy transfer)

COMBUSTION balances
CFD8r
mi mass fraction of specie i in mixture [kg of i]/[kg of mixture]
mi mass concentration of specie [kg of i]/[m3
]
Mass balance of species (for each specie one transport equation)
( ) ( ) ( )
i i i i i
m
t
S
m u m
 

   


Rate of production
of specie i [kg/m3
s]
Production of species is controlled by
Diffusion of reactants (micromixing) – tdiffusion (diffusion time constant)
Chemistry (rate equation for perfectly mixed reactants) – treaction (reaction constant)
Damkohler number
diffusion
reaction
t
Da
t


COMBUSTION enthalpy
CFD8r
Enthalpy balance is written for mixture of all species (result-temperature field)
( ) ( ) ( ) h
S
h hu h
t
 

  


Sum of all reaction enthalpies of all
reactions
h i ri
i
S S h
 

It holds only for reaction without
phase changes h ~ cpT
Energy transport must be solved together with the fluid flow equations (usually
using turbulent models, k-, RSM,…). Special attention must be paid to radiative
energy transport (not discussed here, see e.g. P1-model, DTRM-discrete
transfer radiation,…). For modeling of chemistry and transport of species there
exist many different methods and only one - mixture fraction method will be
discussed in more details.

MIXTURE Fraction method
CFD8r
Bacon

CFD8r
Non-premixed combusion, and assumed fast chemical reactions (paraphrased
as “What is mixed is burned or is at equilibrium”)
mfuel
moxidiser
Flue gases
Mass balance of fuel
Mass balance of oxidant
( ) ( ) ( )
fuel fuel fuel fuel
m m u m S
t
 

  


( ) ( ) ( )
ox ox ox ox
m m u m S
t
 

  


Calculation of fuel and oxidiser
consumption is the most
difficult part. Mixture fraction
method is the way, how to
avoid it
3
of produced fuel
[ ]
kg
s m

Mass fraction of
oxidiser (e.g.air)
Mass fraction of fuel
(e.g.methane)

CFD8r
Stoichiometry
1 kg of fuel + s kg of oxidiser  (1+s) kg of product
and subtracting previous equations
( ) ( ) ( )
( ) ( ) ( )
fuel fuel fuel fuel
ox ox ox ox
s s s s
m m u m S
t
m m u m S
t
 
 

  


  



( ) ( ) ( ) fuel ox
u S S
t
s
 

     


Introducing new variable
fuel ox
sm m
  
This term is ZERO
due to stoichimetry

CFD8r
Mixture fraction f is defined as linear function of  normalized in such a way that
f=0 at oxidising stream and f=1 in the fuel stream
Resulting transport equation for the mixture fraction f is without any source term
( ) ( ) ( )
f fu f
t
 

 


,0
0
1 0 ,1 ,0
fuel ox ox
fuel ox
s
f
s
m m m
m m
 
  
 
   
Mixture fraction is property that is CONSERVED, only dispersed and transported
by convection. f can be interpreted as a concentration of a key element (for
example carbon). And because it was assumed that „what is mixed is burned“ the
information about the carbon concentration at a place x,y,z bears information
about all other participating species.
mox is the mass fraction of
oxidiser at an arbitrary point
x,y,z, while mox,0 at inlet (at
the stream 0)

CFD8r
Knowing f we can calculate mass fraction of fuel and oxidiser at any place x,y,z
,0 ,0
,1 ,0 ,1 ,0
fuel ox ox ox
fue
stoichio
l ox fuel ox
s
f
s s
m m m m
m m m m
 
 
 
For example the mass fraction of fuel is calculated as
ox ,1
(fuel rich region, oxidiser is consumed m =0)
1
1
fuel fu
stoichio
stoich e
io
stoichio
l
f
f m m
f
f
f

  

(fuel lean region)
0 0
stoichio fuel
f
f m
  
The concept can be generalized assuming that chemical reactions are at
equlibrium
f → mi mass fraction of species is calculated from equilibrium constants
(evaluated from Gibbs energies)
At the point x,y,z where
f=fstoichio are all reactants
consumed (therefore
mox=mfuel=0)

CFD8r
Equilibrium depends upon concentration of the key component (upon f) and
temperature. Mixture fraction f undergoes turbulent fluctuations and these
fluctuations are characterized by probability density function p(f). Mean value of
mass fraction, for example the mass fraction of fuel is to be calculated from this
distribution
1
0
( ) ( )
fuel fuel
m m f p f df

Mass fraction corresponding to an
arbitrary value of mixture fraction is
calculated from equlibrium constant
Probability density function, defined in
terms of mean and variance of f
0 fmean 1
p
Frequently used  distribution
1 1
1
1 1
0
(1 )
( )
(1 )
p q
p q
f f
p f
f f df
 
 




Variance of f is calculated from another transport equation
2 2 2 2 2
( ' ) ( ' ) ( ' ) ( ) '
f g t d
f f u f C f C f
t k

   

     



CFD8r
Final remark: In the case, that mfuel is a linear function of f, the mean value of
mass fraction mfuel can be evaluated directly from the mean value of f (and it is
not necessary to identify probability density function p(f), that is to solve the
transport equation for variation of f). Unfortunately the relationship mfuel(f) is
usually highly nonlinear.
1
0
( ) ( ) ( )
fuel fuel fuel
m m f p f df m f
 


COMBUSTION of liquid fuel
CFD8r
mfuel
1
| | ( )
2
D D
du
m F
dt
F c A u v u v


  
 
    
Lagrangian method: trajectories, heating and evaporation of droplets
injected from a nozzle are calculated. Sum of all forces acting to liquid
droplet moving in continuous
fluid (fluid velocity v is calculated
by solution of NS equations)
Relative velocity
(fluid-particle)
Drag force
Drag coefficient cD
depends upon Reynolds number
0.687
5
24 3
(1 Re) Re 5 Oseen
Re 16
24
(1 0.15Re ) Re 800 Schiller Nauman
Re
0.4 1000< Re 3.10 Newton
D
D
D
c
c
c
  
  
 
1 104
105
Re
cD
Newton’s region
cD=0.44
Effect of cloud (c volume
fraction of dispersed phase-gas)
3.7
0 /
D D c
c c 


COMBUSTION of liquid fuel
CFD8r
Evaporation of fuel droplet
Diffusion from droplet surface to gas:
2
2
05 0.33
( ) ( )
6
2 0.6Re
p g dif fg s
dm d D
D Sh D m m
dt dt
Sh Sc

  
  
 
Mass fraction of
fuel at surface
Sherwood
number
Schmidt number =/Ddif
Ranz Marshall correlation for mass transport

MULTIPHASE flows examples
CFD8r
Fluidised bed
reactor or
combustor
Mixer (draft
tube)
Spray
dryer
Annular
flow
Slug flow
Bubble
flow
Flow
boiling
…and others
Hydrotransport,
cyclones, free
surfaces, breakup
of liquid jets,
expanding foams,
aerated reactors,
cavitation, mold
filling…
Phases
Gas-liquid
Gas-solid
Liquid-liquid
visualisation
THERMOPEDIA

MULTIPHASE flows methods
CFD8r
Methods
Lagrange (see liquid fuel burners, suitable for low concentration of particles)
Mixture (not significant difference between phases, e.g. sedimentation)
Euler (the most frequently used technique for any combination of phases)
VOF (Volume Of Fluid) (evolution of continuous interface, e.g. shape of
free surface modeling, moving front of melted solid…)

MULTIPHASE EULER
CFD8r
For each phase q are separately solved
Continuity equation (mass balance of phase)
Momentum balance (each phase is moving with its own velocity,
only pressure is common for all phases)
( ) ( )
q q q q q pq
p
v m
t
   

 




Volumetric
fraction of phase
q
Velocity of
phase q
Mass transfer from
phase p to phase q
( ) ( ) ( )
q q q q q q q q q q pq
p
v v v p R
t
      

    


 
   
Interphas
e forces
Stresses are calculated in the
same way like in one phase
flows

MULTIPHASE EULER
CFD8r
Specific semiemprical correlations describe
interaction terms
Mass transfer
for example Ranz Marschall correlation for Sh=2+…
Momentum exchange ( )
pq pq q p
R k v v
 
  
Special models for kpq are
available for liquid-liquid, liquid-
solid, and also for solid-solid
combinations

MULTIPHASE MIXTURE method
CFD8
Mixture model solves in principle one-phase flow with mean
density m , mean velocity vm
Continutity equation for mixture
Momentum balance for mixture (with corrections to drift velocities)
Volumetric fraction of secondary phase (p)
( ) 0
m
m m
v
t



 


, ,
,
( ) ( ) ( ( ( ) ) ( )
T
m m m m m m m m p p dr p dr p
p
dr p p m
v v v p v v v v
t
v v v
    

       

 

      
  
,
( ) ( ) ( )
p p p p m p p dr p
v v
t
     

  

 
Drift velocities are evaluated from algebraic models
(mixture acceleration determines for example centrifugal
forces applied to phases with different density)

MULTIPHASE VOF
CFD8
Evolution of clearly discernible interface between
immiscible fluids (examples: jet breakup, motion of large bubbles, free surface
flow)
There exist many different methods in this category, Level set method,
Marker and cell, Lagrangian method tracking motion of particles at interface.
=0
=0
=0
=0
=1
=1 =1 =1

 

Fluent
Donor acceptor
Geometric reconstruction
0
u
t



 
 


Dissadvantage: initially sharp
interface is blurred due to
numerical diffusion

RADIATION
CFD8
Heat flow (W) between gas and wall
 .
4
4
w
g
S
g T
A
T
S
Q 
 


Emissivity of gas
corresponding to
temperature of gas Ts
Absorptivity of gas
corresponding to wall
temperature Tw
Hottel’s diagram for
emissivity of CO2 and H2O
as a function of temperature and pL
(pressure x length)
 
    
,
1
4
2
2
2 10
8
,
3
1
160
8
10
T
p
L
p
p
g
O
H
CO
O
H
e
T









Kirchhoff’s law (=a)
Emissivity=Absorptivity
at the same wavelength

RADIATION Fluent models
CFD8
Radiation models are selected according to optical thickness of media (flue gas)
a.L 3.5
V
L
S

aL<1
DTRM (discrete transf.radiation modelling)
DO (discrete ordinates)
1<aL<3
P-1 model (transport equation for radiation temperature)
3<aL
Roseland model (simplified P-1 model)
4
( )
s
dI a T
a I
dt



  
Absorption and
scatter

computational work for compressible flow

More Related Content

Similar to computational work for compressible flow (20)

More from AzeemKhan17786 (7)

Recently uploaded (20)

computational work for compressible flow