Download Computational models for prediction of fire behaviour

Computational Models for
Prediction of Fire Behaviour
Dr. Andrej Horvat
Intelligent Fluid Solutions Ltd.
Ljubljana, Slovenia
17 January, 2008
Contact information
IFS
Andrej Horvat
Intelligent Fluid Solution Ltd.
127 Crookston Road, London, SE9 1YF, United Kingdom
Tel./Fax: +44 (0)1235 819 729
Mobile: +44 (0)78 33 55 63 73
E-mail: [email protected]
Web: www.intelligentfluidsolutions.co.uk
2
Personal information
IFS

1995, Dipl. -Ing. Mech. Eng. (Process Tech.)
University of Maribor

1998, M.Sc. Nuclear Eng.
University of Ljubljana

2001, Ph.D. Nuclear Eng.
University of Ljubljana

2002, M.Sc. Mech. Eng. (Fluid Mechanics & Heat Transfer)
University of California, Los Angeles
3
Personal information
IFS
More than 10 years of intensive CFD related experience:

R&D of numerical methods and their implementation
(convection schemes, LES methods, semi-analytical methods,
Reynolds Stress models)

Design analysis
(large heat exchangers, small heat sinks, burners, drilling equip.,
flash furnaces, submersibles)

Fire prediction and suppression
(backdraft, flashover, marine environment, gas releases,
determination of evacuation criteria)

Safety calculations for nuclear and oil industry
(water hammer, PSA methods, severe accidents scenarios, pollution
dispersion)
4
Personal information
IFS
As well as CFD, experiences also in:

Experimental methods

QA procedures

Standardisation and technical regulations

Commercialisation of technical expertise and software
products
5
Contents

IFS
Overview of fluid dynamics transport equations
- transport of mass, momentum, energy and composition
- influence of convection, diffusion, volumetric (buoyancy) force
- transport equation for thermal radiation

Averaging and simplification of transport equations
- spatial averaging
- time averaging
- influence of averaging on zone and field models

Zone models
- basics of zone models (1 and 2 zone models)
- advantages and disadvantages
6
Contents

Field models
-

IFS
numerical mesh and discretisation of transport equations
turbulence models (k-epsilon, k-omega, Reynolds stress, LES)
combustion models (mixture fraction, eddy dissipation, flamelet)
thermal radiation models (discrete transfer, Monte Carlo)
examples of use
Conclusions
- software packages

Examples
- diffusion flame
- fire in an enclosure
- fire in a tunnel
7
Some basic thoughts …
IFS

Today, CFD methods are well established tools that help in design,
prototyping, testing and analysis

The motivation for development of modelling methods (not only CFD)
is to reduce cost and time of product development, and to improve
efficiency and safety of existing products and installations

Verification and validation of modelling approaches by comparing
computed results with experimental data are necessary

Nevertheless, in some cases CFD is the only viable research and
design tool (e.g. hypersonic flows in rarefied atmosphere)
8
IFS
Overview of fluid dynamics
transport equations
9
Transport equations
IFS
The continuum assumption
- A control volume has to contain a large number of particles (atoms or
molecules): Knudsen number << 1.0
- At equal distribution of particles (atoms or molecules) flow quantities
remain unchanged despite of changes in location and size of a control
10
volume
Transport equations
IFS
 Eulerian and Lagrangian description
 Eulerian description – transport equations for mass, momentum
and energy are written for a (stationary) control volume
 Lagrangian description – transport equations for mass,
momentum and energy are written for a moving material particle
11
IFS
Transport equations
 Transport of mass and composition
dm d

  dV
dt dt MR
dm j
dt

d
   j dV
dt MR
Transport of momentum
 Transport of energy
12
Transport equations
IFS
 Majority of the numerical modelling in fluid mechanics is
based on Eulerian formulation of transport equations
 Using the Eulerian formulation, each physical quantity is
described as a mathematical field. Therefore, these
models are also named field models
 Lagrangian formulation is basis for modelling of particle
dynamics: bubbles, droplets (sprinklers), solid particles
(dust) etc.
13
Transport equations
IFS
Droplets trajectories from sprinklers (left), gas temperature
field during fire suppression (right)
14
IFS
Transport equations
 Eulerian formulation of mass transport equation
integral form
differential (weak) form
The equation also appears in the following forms
change of mass
in a control vol.
flux difference
(convection)
non-conservative form
~0
in incompressible 15
fluid flow
IFS
Transport equations
 Eulerian formulation of mass fraction transport equation
(general form)
d
   j dV  MR ni qi dS
dt MR
 t   j    i   vi  j    i qi
change of mass of a
flux
component in a
difference
control vol.
(convection)
diffusive
mass flow
16
Transport equations
IFS
 Eulerian formulation of momentum equation
change of momentum
volumetric pressure viscous
force
in a control vol.
force
flux difference force
(diffusion)
(convection)
17
Transport equations
IFS
 Eulerian formulation of energy transport equation
change of internal energy
flux
deformation diffusive heat
in a control vol.
difference
work
flow
(convection)
18
IFS
Transport equations
 The following physical laws and terms also need to be
included
diffusive terms - Newton's viscosity law
flux is a linear
- Fourier's law of heat conduction
function of a gradient
- Fick's law of mass transfer

- Sources and sinks due to thermal radiation, chemical
reactions etc.
19
Transport equations
IFS
 Transport of mass and composition
 t    i vi   M
 t  j    i vi  j    i D i  j   M j
 Transport of momentum
 t v j    i vi v j    j p  2 i Sij   g j  F j
Sij 
1
 j vi   i v j 
2
 Transport of energy
 t h   i vi h   i  iT   Q
20
IFS
Transport equations
 Lagrangian formulation is simpler
- equation of the particle location

dx 
u
dt
- mass conservation eq. for a particle
dm
M
dt
- momentum conservation eq. for a particle



du 
m
 FD  FL  FV
dt
drag
lift
volumetric
forces
21
IFS
Transport equations
- thermal energy conservation eq. for a particle
mc p
dT
 QC  QL  QR
dt
convection
latent
heat
thermal
radiation
Conservation equations of Lagrangian model need to be
solved for each representative particle
22
IFS
Transport equations
Thermal radiation
s
I(s)
I(s+ds)
dA
Ie
Is
K
dI
 K a  K s I  K a I e  s  I s P'  d'
ds
4 4 
change of
radiation
intensity
absorption emission
and
out-scattering
in-scattering
23
IFS
Transport equations
 Thermal radiation
Equations describing thermal radiation are much more
complicated
- spectral dependency of material properties
- angular (directional) dependence of the radiation transport
dI  
K
 K a  K s I    K a I e  s
ds
4
change of
radiation
intensity
absorption emission
and
out-scattering
 I ' P '  d'
s

4
in-scattering
24
IFS
Averaging and simplification
of transport equations
25
Averaging and simplification of
transport equations
IFS
 The presented set of transport equations is analytically
unsolvable for the majority of cases
 Success of a numerical solving procedure is based on
density of the numerical grid, and in transient cases, also
on the size of the integration time-step
 Averaging and simplification of transport equations help
(and improve) solving the system of equations:
- derivation of averaged transport equations for turbulent
flow simulation
- derivation of integral (zone) models
26
IFS
Averaging and simplification of
transport equations
Averaging and filtering
, vi , p, h
,
w
The largest flow structures can occupy the whole flow
field, whereas the smallest vortices have the size of
Kolmogorov scale

  
3

1
4
u  
1
4
    
1
2
27
Averaging and simplification of
transport equations
IFS
Kolmogorov scale is (for most cases) too small to be
captured with a numerical grid
Therefore, the transport equations have to be filtered
(averaged) over:
- spatial interval  Large Eddy Simulation (LES)
methods

h  x ,t    G x    h ,t  d  h x ,t   h  x ,t   h'  x ,t 

- time interval  k-epsilon model, SST model,
Reynolds stress models

h  x ,t    G t   h x ,  d  h x ,t   h x ,t   h x ,t 

28
IFS
Averaging and simplification of
transport equations
Transport equation variables can be decomposed onto
a filtered (averaged) part and a residual (fluctuation)
    '
p  p  p'
vi  vi   vi*  v~i  vi*
~
i  i  *
~
hi  h  h*
Filtered (averaged) transport equations
  

~
~
 t   j   i  v~i  j  M j   i j 
 t    j ( v~j )  M
t  v~j   i  v~i v~j    j p  2i Sij   g j  Fj  i ij 
  



~
~
~
t  h  i v~i h  i iT  Q  i i 
turbulent heat fluxes
turbulent mass
fluxes
sources and sinks
represent a separate
problem and require
additional models
- turbulent stresses
- Reynolds stresses
- subgrid stresses
29
Averaging and simplification of
transport equations
 Turbulent stresses
IFS
  ji  v j vi   v~j v~i   v*j vi*
 Transport equation
 t   ji    k  v~k  ji  

  k p' v*j ik  p' vi*  jk   vi* v*j v*k  v*jTik '  vi*T jk '

 p' ( j vi*   i v*j )

   ik  k v~j    jk  k v~i  F j vi  F j v~i  Fi v j  Fi v~j
 Tik '  v  T jk '  v
*
k j
*
k i
 v*j  k  p ik  Tik   vi*  k  p  jk  T jk 

higher order
product
turbulence
generation
turbulence
dissipation
- the equation is not solvable due to the higher order product
- all turbulence models include at least some of the terms of this
equation (at least the generation and the dissipation term)
30
Averaging and simplification of
transport equations
IFS
 Turbulent heat and mass fluxes
~
  j  v j h   v~j h
 Transport equation
 t  j    k  v~k  j  

  k p' h*  jk  v*k v*j h*  v*j qk '  h*Tkj '

 

~
~
~
 vk vj  k h  vk h  k v~j  F j h  F j h  Q h  Q h
 qk  k v*j  Tkj  k h*

 v*j h*
higher order product
generation
dissipation
 h*  k  p jk  Tkj   v*j  k qk
- the equation is not solvable due to the higher order product
- most of the turbulence models do not take into account the
equation
31
IFS
Averaging and simplification of
transport equations
 Turbulent heat fluxes due to thermal radiation
- little is known and published on the subject
- majority of models do not include this contribution
- radiation heat flow due to turbulence
Qs ~ T ' 4
32
Averaging and simplification of
transport equations
IFS
Buoyancy induced flow over a heat source (Gr=10e10);
inert model of a fire
33
Averaging and simplification of
transport equations
IFS
(a)
LES model; instantaneous temperature field
34
Averaging and simplification of
transport equations
(a)
a)
IFS
b)
Temperature field comparison:
a) steady-state RANS model, b) averaged LES model results
35
Averaging and simplification of
transport equations
(a)
IFS
a)
b)
Comparison of instantaneous mass fraction in a gravity
current : a) transient RANS model, b) LES model
36
Averaging and simplification of
transport equations
IFS
 Additional simplifications
- flow can be modelled as a steady-state case  the
solution is a result of force, energy and mass flow balance
taking into consideration sources and sinks
- fire can be modelled as a simple heat source  inert
models; do not need to solve transport equations for
composition
- thermal radiation heat transfer is modelled as a simple sink
of thermal energy  FDS takes 35% of thermal energy
- control volumes can be so large that continuity of flow
properties is not preserved  zone models
37
IFS
Zone models
38
Zone models
IFS
 Basics
- theoretical base of zone model is conservation of mass
and energy in a space separated onto zones
- thermodynamic conditions in a zone are constant; in
fields models the conditions are constant in a control
volume
- zone models take into account released heat due to
combustion of flammable materials, buoyant flows as a
consequence of fire, mass flow, smoke dynamics and
gas temperature
- zone models are based on certain empirical assumptions
- in general, they can be divided onto one- and two-zone
models
39
IFS
Zone models
 One-zone and two-zone models
- one-zone models can be used only for assessment of
a fully developed fire after flashover
- in such conditions, a valid approximation is that the gas
temperature, density, internal energy and pressure are
(more or less) constant across the room
Qw (konv+rad)
Qout (konv+rad)
Qin (konv)
pg , Tg , mg , vg
mout
mf , H f
min
40
IFS
Zone models
 One-zone and two-zone models
- two-zone models can be used for evaluation of a
localised fire before flashover
- a room is separated onto different zone, most often
onto an upper and lower zone, a fire and buoyant flow
of gases above the fire
- conditions are uniform and constant in each zone
Qw (konv+rad)
mU,out
mL,out
mf , Hf
Qin (konv)
zgornja cona QU,out (konv+rad)
pU,g , TU,g , mU,g , vU,g
spodnja cona
pL,g , TL,g , mL,g , vL,g
mL,in
QL,out (konv+rad)
41
Zone models
IFS
 Advantages and disadvantages of zone models
- in zone models, ordinary differential equations describe
the conditions  more easily solvable equations
- because of small number of zones, the models are fast
- simple setup of different arrangement of spaces as well
as of size and location of openings
- these models can be used only in the frame of theoretical
assumptions that they are based on
- they cannot be used to obtain a detailed picture of flow
and thermal conditions
- these models are limited to the geometrical
arrangements that they can describe
42
IFS
Field models
43
Field models
IFS
 Numerical grid and discretisation of transport equations
- analytical solutions of transport equations are known only
for few very simple cases
- for most real world cases, one needs to use numerical
methods and algorithms, which transform partial
differential equations to a series of algebraic equations
- each discrete point in time and space corresponds to an
equation, which connects a grid point with its neighbours
- The process is called discretisation. The following
methods are used: finite difference method, finite volume
method, finite element method and boundary element
method (and different hybrid methods)
44
IFS
Field models
Numerical grid and discretisation of transport equations
- simple example of discretisation
ui 1  ui
x 0
x
 x u  lim
ui 1  2ui  ui 1
x 0
x 2
 x  x u  lim
- many different discretisation schemes exist; they can be
divided onto conservative and non-conservative schemes
(linked to the discrete form of the convection term)
ui 1ui 1  ui ui
x 0
x
 x u u   lim
u  x u   lim ui 1 2
x 0
ui 1  ui
x
45
Field models
IFS
 Numerical grid and discretisation of transport equations
- non-conservative schemes represent a linear form and
therefore they are more stable and numerically better
manageable
- non-conservative schemes do not conserve transported
quantities, which can lead to time-shift of a numerical
solution
time-shift due to a
non-conservative
scheme
46
IFS
Field models
 Numerical grid and discretisation of transport equations
- quality of numerical discretisation is defined with
discrepancy between a numerical approximation and an
analytical solution
- error is closely link to the order of discretisation
x 2
u x  x   u x   x  xu x  
 x  xu x   .......
2
u x  x   u x 
x
  xu x  
 x  xu x   . ......
x
2
1st order truncation (~x)
47
Field models
IFS
 Numerical grid and discretisation of transport equations
- higher order methods lead to lower truncation errors, but
more neighbouring nodes are needed to define a
derivative for a discretised transport equation
- two types of numerical error: dissipation and dispersion
48
Field models
IFS
 Numerical grid and discretisation of transport equations
- during the discretisation process most of the attention
goes to the convection terms
- the 1st order methods have dissipative truncation error,
whereas the 2nd order methods introduce numerical
dispersion
- today's hybrid methods are a combination of 1st and 2nd
order accurate schemes. These methods switch
automatically from a 2nd order to a 1st order scheme near
discontinuities to damp oscillations (TVD schemes)
- there are also higher order schemes (ENO, WENO etc.)
but their use is limited on structured numerical meshes
49
IFS
Field models
 Numerical grid and discretisation of transport equations
- connection matrix and location of neighbouring grid
nodes defines two types of numerical grid: structured and
unstructured numerical grid
i-1, j+1
i, j+1
i+1, j+1
k+2
k+1
k+6
i-1, j
i, j
i+1, j
k+9
k
k+3
i-1, j-1
i, j-1
i+1, j-1
k+7
k+5
k+4
structured grid
unstructured grid
50
Field models
IFS
 Numerical grid and discretisation of transport equations
- unstructured grids raise a possibility of arbitrary
orientation and form of control volumes and therefore
offer larger geometrical flexibility
- all modern simulation packages (CFX, Fluent, Star-CD)
are based on the unstructured grid arrangement
51
Field models
IFS
 Numerical grid and discretisation of transport equations
- flame above a burner
- automatic refinement of
unstructured numerical
grid
- start with 44,800 control
volumes; 3 stages of
refinement which leads to
150,000 cont. volumes
- refinement criteria velocity and temperature
gradients
52
IFS
Turbulence
models
53
Turbulence models
IFS
laminar flow
transitional flow
turbulent flow
van Dyke, 1965
54
Turbulence models
IFS
Turbulence models introduce additional (physically related)
diffusion to a numerical simulation
This enables :
- RANS models to use a larger time step (Δt >> Kolmogorov
time scale) or even steady-state simulation
- LES models to use less dense (smaller) numerical grid
(Δx > Kolmogorov length scale)
The selection of the turbulence model influences the
distribution of the simulated flow fundamentally and hence
that of the flow variables (velocity, temperature, heat flow,
composition etc)
55
Turbulence models
IFS
 In general 2 kinds of averaging (filtering) exist, which
leads to 2 families of turbulence models:
- filtering over a spatial interval  Large Eddy Simulation
(LES) models
- filtering over a time interval  Reynolds Averaged
Navier-Stokes (RANS) models: k-epsilon model, SST
model, Reynolds Stress models etc
 For RANS models, size of the averaging time interval is
not known or given (statistical average of experimental
data)
 For LES models, size of the filter or the spatial averaging
interval is a basic input parameter (in most case it is
equal to grid spacing)
56
IFS
Turbulence models

Reynolds Averaged Navier-Stokes (RANS) models
For two-equation models (e.g. k-epsilon, k-omega or
SST), 2 additional transport equations need to be solved:
- for kinetic energy of turbulent fluctuations
k  1 2  ii
- for dissipation of turbulent fluctuations
    ( j vi*  j vi* )
or
- for frequency of turbulent fluctuations
~  k
57
IFS
Turbulence models

Reynolds Averaged Navier-Stokes (RANS) models
k-epsilon model



 t k    j v~j k   P  G   j  (  t )  j k   
k




t

2
~
 t     j v j    C1 P  C3 max(G ,0)    j  (  )  j    C2 
k

k


convection
production
(source) term
production (source) term
diffusion
destruction
(sink) term
2
P  2 t Sij  j v~i   l v~l  k  3 t  l v~l 
3
G
t
g j  j  
Prt
58
IFS
Turbulence models

Reynolds Averaged Navier-Stokes (RANS) models
- model parameters are usually defined from experimental
data e.g. dissipation of grid generated turbulence or flow
in a channel
C
C
C
C

Pr


1
0.09
1.44
2
1.92
3
1.0
k
1.0

1.3
t
0.9
- from the calculated values of k in , eddy viscosity is
defined as
2
 t  C ρ
k

- from eddy viscosity, Reynolds stresses, turbulent heat and
mass fluxes are obtained
1
2
  ij    ll  ji  2 t Sij   t ( l v~l )  ji
3
3
 j  
t ~
 jh
Prt
j  
t ~
 j
Sct
59
Turbulence models

IFS
Reynolds Averaged Navier-Stokes (RANS) models
- transport equation for k is derived directly from the
transport equations for Reynolds stresses  ij
- transport equation for  is empirical
- these equations have a common form: source and sink
term, convection and diffusion term
- the rest of the terms are model specific and they can be
numerically and computationally very demanding
- definition and implementation of boundary conditions is
different even for the same turbulence model between
different software vendors
60
IFS
Turbulence models

Reynolds Averaged Navier-Stokes (RANS) models
For Reynolds Stress (RS) model, 7 additional transport
equations need to be solved:
- for 6 components of Reynolds stress tensor
 xx ,  xy ,  xz ,  yy ,  yz ,  zz
- for dissipation of turbulent fluctuations
    ( j vi*  j vi* )
or
- for frequency of turbulent fluctuations
~  k
61
Turbulence models

IFS
Reynolds Averaged Navier-Stokes (RANS) models
Reynolds stress model

 2
2
k2
~
 t   ij    l  vl  ij   Pij   ij   l  (   Cs 
)  l  ij   ij
3


 3


t

2
~
 t     l  vl   C1 P   l  (  )  l    C 2 
k

k


convection
generation
(source) term
generation
diffusion term
(source) term
pressure-velocity
fluctuation term
destruction
(sink) term
Pij     ik   k u~ j     jk  k u~i
62
Turbulence models

IFS
Reynolds Averaged Navier-Stokes (RANS) models
- Reynolds stress models calculate turbulent stresses
directly  introduction of eddy viscosity is not needed
- includes the pressure-velocity fluctuation term
 
2
1



 ij   C1   ij  kij   C2  Pij  Pll ij 
k 
3
3



- describes anisotropic turbulent behaviour
- computationally more demanding than the two-equation
models
- due to higher order terms (linear and non-linear), RS
models are numerically more complex (convergence)
63
Turbulence models

IFS
Large Eddy Simulation (LES) models
- Large Eddy Simulation (LES) models are based on spatial
filtering (averaging)
- many different forms of the filter exist, but most common
is "top hat" filter (simple geometrical averaging)
- the size of the filter is based on grid node spacing
Basic assumption of LES methodology:
Size of the used filter is so small that the averaged flow
structures do no influence large structures, which contain
most of the energy.
These small structures are being deformed, disintegrated
onto even smaller structures until they do not dissipate
due to viscosity (kinetic energy  thermal energy).
64
IFS
Turbulence models

Large Eddy Simulation (LES) models
- required size of flow structures for LES modelling
1 k
l  lEI ~  
6 
32
- these structures are in the turbulence inertial subrange
and they are isotropic
- on this level, turbulence production is of the same size as
turbulence dissipation


 ik  k v~i  Fi vi  Fi v~i  
65
IFS
Turbulence models

Large Eddy Simulation (LES) models
- eddy (turbulent) viscosity is defined as
 t ~ l 4 / 31 / 3
where
l ~ Cs 
grid spacing
- using the definition of turbulent (subgrid) stresses
2
2
  ji   k  ji  2 t Sij   t ( l v~l )  ji
3
3
and turbulent fluxes
 j  
t ~
 jh
Prt
the expression for turbulent viscosity can be written as
t  Cs   2S ji Sij  G 
2
1/ 2
where the contribution
due to buoyancy is
~
gi  i h
G~
~
Prt h
66
Turbulence models

IFS
Large Eddy Simulation (LES) models
- presented Smagorinsky model is the simplest from LES
models
- it requires knowledge of empirical parameter Cs, which is
not constant for all flow conditions
- newer, dynamic LES models calculates Cs locally - the
procedure demands introduction of the secondary filter
- LES models demand much denser (larger) numerical grid
- they are used for transient simulations
- to obtain average flow characteristics, we need to perform
statistical averaging over the simulated time interval
67
IFS
Turbulence models

Comparison of turbulence models
10
10
k-e (Gr = 10 )
k-e N&B (Gr = 10 10 )
10
RNG k-e (Gr = 10 )
10
S S T (Gr = 10 )
10
S S G (Gr = 10 )
10
LES (Gr = 10 )
Rous e e t al. (1952)
S habbir and Ge orge (1994)
8
7
6
5
4
k-e (Gr = 10 )
k-e N&B (Gr = 10 10 )
10
RNG k-e (Gr = 10 )
10
S S T (Gr = 10 )
10
S S G (Gr = 10 )
10
LES (Gr = 10 )
Rous e e t al. (1952)
S habbir and Ge orge (1994)
10 -1
b c (R 5 /F 02 )1/3
w c (R/F0 )1/3
3
2
1
10 -2
25
50
z/R
a)
75
100
25
50
75
100
z/R
b)
Buoyant flow over a heat source: a) velocity, b) temperature*
68
IFS
Combustion
models
69
IFS
Combustion models
Chen et al., 1988
Grinstein,
Kailasanath, 1992
70
Combustion models
IFS
 Combustion can be modelled with heat sources
- information on chemical composition is lost
- thermal loading is usually under-estimated
 Combustion modelling contains
- solving transport equations for composition
- chemical balance equation
- reaction rate model
 Modelling approach dictates the number of additional
transport equations required
71
IFS
Combustion models
 Modelling of composition requires solving n-1 transport
equations for mixture components – mass or molar
(volume) fractions
  
~
~
 t   j   i  v~i  j   i    t
  Sc Sct
  

 ~ 
 i  j   M j



 Chemical balance equation can be written as
' A A  ' B B  'C C .....  " A A  " B B  "C C .....
or
N
N
 ' I   " I
I  A ,B ,C....
I
I  A ,B ,C....
I
72
IFS
Combustion models
 Reaction source term is defined as
M j  W j " j ' j R
or for multiple
reactions
M j  W j  "k , j ' k , j Rk
k
where R or Rk is a reaction rate
 Reaction rate is determined using different models
- Constant burning (reaction) velocity
- Eddy break-up model and Eddy dissipation model
- Finite rate chemistry model
- Flamelet model
- Burning velocity model
73
IFS
Combustion models
 Constant burning velocity sL
sF 
c
sL
h
speed of flame front propagation is
larger due to expansion
- values are experimentally determined for ideal conditions
- limits due reaction kinetics and fluid mechanics are not
taken into account
- source/sink in mass fraction transport equation
- source/sink in energy transport equation
M j ~  f sL l
Q ~ M j hc
- expressions for sL usually include additional models
74
IFS
Combustion models
 Eddy break-up model and Eddy dissipation model
- is a well established model
- based on the assumption that the reaction is much faster
than the transport processes in flow
- reaction rate depends on mixing rate of reactants in
turbulent flow
s ~ k
L
- Eddy break-up model reaction rate
R ~

' 2f
k
- Eddy dissipation model reaction rate
~
~

~ 

p
o
R  C A  min   f , ,CB
k
s
1  s 




75
Combustion models
IFS
 Eddy break-up model and Eddy dissipation model
- typical values of model coefficients CA = 4 in CB = 0.5
- the model can be used for simple reactions (one- and
two-step combustion)
- in general, it cannot be used for prediction of products of
complex chemical processes (NO, CO, SOx, etc)
- the use can be extended by adding different reaction rate
limiters → extinction due to turbulence, low temperature,
chemical time scale etc.
76
Combustion models
IFS
 Backdraft simulation
77
Combustion models
IFS
 Finite rate chemistry model
- it is applicable when a chemical reaction rate is slow or
comparable with turbulent mixing
- reaction kinetics must be known
~
E 
~ ' I
R  AT  exp  ~a   
I
 RT  I  A ,B ,C ...
- for each additional component, the component molar
concentration I needs to be multiplied with the product
- for each additional reaction the same expression is
added
78
Combustion models
IFS
 Finite rate chemistry model
- the model is numerically demanding due to exponential
terms
- often the model is used in combination with the Eddy
dissipation model
79
Combustion models
IFS
 Flamelet model
- describes interaction of reaction kinetics with turbulent
structures for a fast reaction (high Damköhler number)
- basic assumption is that combustion is taking place in
thin sheets - flamelets
- turbulent flame is an ensemble of laminar flamelets
- the model gives a detailed picture of chemical
composition - resolution of small length and time scales
of the flow is not needed
- the model is also known as "Mixed-is-burnt" - large
difference between various implementations of the
model
80
IFS
Combustion models
 Flamelet model
- the model is only applicable for two-feed systems
(fuel and oxidiser)
- it is based on definition of a mixture fraction
Z kg/s fuel
A
1-Z kg/s oxidiser
B
mešalni
proces
Z  A  1  Z B  M
or
1 kg/s mixture
M
Z
M  B
 A  B
Z is 1 in a fuel stream, 0 in an oxidiser stream
81
IFS
Combustion models
 Flamelet model
- conserved property often used in the mixture fraction
definition    f  o i
- for a simple chemical reaction
1kg fuel  i kg oxidiser  1  i  kg products
the stoihiometric ratio is
f ST  1 (1  i)
82
Combustion models
IFS
 Flamelet model
- Shvab-Zel'dovich variable
   

 f  o    o 
i   i B
Z
 f A   io 
 B
- the conditions in vicinity of flamelets are described
with the respect to Z; Z=Zst is a surface with the
stoihiometric conditions
- transport equations are rewritten with Z dependencies;
conditions are one-dimensional ξ(Z) , T(Z) etc.
- for limited cases, the simplified equations can be solve
analytically
83
IFS
Combustion models
 Flamelet model
- numerical implementations of the model differ significantly
- for turbulent flow, we need to solve an additional transport
equation for mixture fraction Z
  
~
~
 t  Z   i  v~i Z   i    t
  Sc Sct
  

 ~
 i Z 



- and a transport equation for variation of mixture fraction Z"

 ~'' 2 
 ~ ~'' 2 

 t   Z    i   ui Z    i   D  t
Sct





~
 ~'' 2 
t
 '' 2
~2
  i Z   2
 i Z   C Z
Sc
k
t


 
84
IFS
Combustion models
 Flamelet model
- composition is calculated from preloaded libraries
1
~   Z  PDF Z  dZ

j
 j
0
1
~   Z ,  PDF Z  PDF   dZ d

j
 j
0 0
these PDFs are tabulated for different fuel,
oxidiser, pressure and temperature
-  is the scalar dissipation rate
- it increases with stresses (stretching of a flame front)
and decreases with diffusion.
- at critical value of  flame extinguishes
85
Combustion models
IFS
 Flamelet model
Ferreira, Schlatter, 1995
86
Combustion models
IFS
 Burning velocity model
- it is used for pre-mixed or partially pre-mixed combustion
- contains:
a) model of the reaction progress:
Burning Velocity Model (BVM) or Turbulent Flame Closure (TFC)
b) model for composition of the reacting mixture:
Flamelet model
- definition of the reaction progress variable c, where c = 0
corresponds to the fresh mixture, and c = 1 to combustion
products
~  1  c~ 
~
~~

j
j ,svezi  c  j ,zgoreli
87
IFS
Combustion models
 Burning velocity model
Preheating
T, 
j

Reaction layer O()
Oxidation layer
Temperature
Oxidiser
Products
Fuel
x
Flame location
lF  0.1 ... 0.5 mm
l  0.01 mm
88
IFS
Combustion models
 Burning velocity model
- additional transport equation for the reaction progress
variable


~
~
~
 t  c    i  vi c    i    D  t
Sct

 ~
 i c   M

 
source term
- source/sink due to combustion is defined as
M   freshST c~
modelling of the turbulent burning velocity
- better results by solving the transport equation for
weighted reaction progress variable ~ ~ ~
F  Z 1  c 
89
Combustion models
IFS
 Burning velocity model
- it is suitable for a single step reaction
- applicable for fast chemistry conditions (Da >> 1)
- a thin reaction zone assumption
- fresh gases and products need to be separated
90
IFS
Thermal radiation
models
91
IFS
Thermal radiation
 It is a very important heat transfer mechanism during
fire
 In fire simulations, thermal radiation should not be
neglected
 The simplest approach is to reduce the heat release
rate of a fire (35% reduction in FDS)
 Modelling of thermal radiation - solving transport
equation for radiation intensity
dI  
K
 K a  K s I    K a I e  s
ds
4
change of
intensity
absorption emission
and scattering
 I ' P '  d'
s

4
in-scattering
92
IFS
Thermal radiation
 Radiation intensity is used for definition of a source/sink
in the energy transport equation and radiation wall heat
fluxes
 Energy spectrum of blackbody radiation
22
n 2 h
E T   I  T   2
[Wm -2 Hz -1 ]
c exph k BT   1
 - frequency
c - speed of light
n - refraction index
h - Planck's constant
kB - Boltzmann's constant

integration over
the whole spectrum
E T   n T   E T  d [Wm -2 ]
2
4
0
93
IFS
Thermal radiation
 Models
- Rosseland (primitive model, for optical thick systems, additional
diffusion term in the energy transport equation)
q  qd  qr  kd  kr  iT
16n 2T 3
kr  
3
- P1 (strongly simplified radiation transport equation - solution of an
additional Laplace equation is required)
G   I   d
4
- Discrete Transfer (modern deterministic model,
assumes isotropic scattering, reasonably
homogeneous properties)
- Monte Carlo (modern statistical model, computationally
demanding)
94
IFS
Thermal radiation
 Discrete Transfer
- modern deterministic model
- assumes isotropic scattering, homogeneous gas properties
- each wall cell works as a radiating surface that emits rays
through the surrounding space (separated onto multiple solid
angles)
- radiation intensity is integrated along each ray between the
walls of the simulation domain


I  (r , s)  I 0 e - K a  K s s  I e 1  e  K a s  K s I 
qrad
, j   I  r , s  cos  j cos  d
4
- source/sink in the energy transport equation
Q rad  i qirad
95
Thermal radiation
IFS
 Monte Carlo
- it assumes that the radiation intensity is proportional to
(differential angular) flux of photons
- radiation field can be modelled as a "photon gas"
- absorption constant Ka is the probability per unit length of
photon absorption at a given frequency 
- average radiation intensity I is proportional to the photon
travelling distance in a unit volume and time
- radiation heat flux qrad is proportional to the number of
photon incidents on the surface in a unit time
- accuracy of the numerical simulation depends on the
number of used "photons"
96
Thermal radiation
IFS
 These radiation methods can be used:
- for averaged radiation spectrum - grey gas
- for gas mixture, which can be separated onto multiple
grey gases (such grey gas is just a modelling concept)
- for individual frequency bands; physical parameters are
very different for each band
97
Thermal radiation
IFS
 Flashover simulation
98
IFS
Conclusions
99
Conclusions
IFS
 The seminar gave a short (but demanding) overview of
fluid mechanics and heat transfer theory that is relevant
for fire simulations
 All current commercial CFD software packages (ANSYSCFX, ANSYS-Fluent, Star-CD, Flow3D, CFDRC, AVL Fire)
contain most of the shown models and methods:
- they are based on the finite volume or the finite element
method and they use transport equations in their
conservative form
- numerical grid is unstructured for greater geometrical
flexibility
- open-source computational packages exist and are freely
accessible (FDS, OpenFoam, SmartFire, Sophie)
100
Conclusions
IFS
 I would like to remind you to the project "Methodology for
selection and use of fire models in preparation of fire
safety studies, and for intervention groups", sponsored by
Ministry of Defence, R. of Slovenia, which contains an
overview of functionalities that are offered in commercial
software products
101
Acknowledgement
IFS
 I would like to thank to our hosts and especially to Aleš Jug
 I would like to thank to ANSYS Europe Ltd. (UK) who
permitted access to some of the graphical material
102