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									 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: andrej.horvat@intelligentfluidsolutions.co.uk
 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                                                    IFS
   Field models
    -   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
   volume                                                                 10
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
Transport equations                                   IFS
 Transport of mass and composition

     dm d                  dm j
            dV
     dt dt MR               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
Transport equations                                        IFS
 Eulerian formulation of mass transport equation




          integral form                 differential (weak) form

The equation also appears in the following forms




                    flux difference
change of mass
                    (convection)
in a control vol.
                          non-conservative form                        ~0
                                                       in incompressible 15
                                                               fluid flow
Transport equations                                                    IFS
 Eulerian formulation of mass fraction transport equation
  (general form)

                       d
                                        
                             j dV  MRni qi dS
                       dt MR



                       t   j    i   vi  j    i qi



       change of mass of a      flux                       diffusive
       component in a       difference                     mass flow
       control vol.        (convection)



                                                                             16
Transport equations                                  IFS
 Eulerian formulation of momentum equation




  change of momentum               volumetric pressure viscous
  in a control vol. flux difference force       force    force
                     (convection)                     (diffusion)



                                                                    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
Transport equations                            IFS
 The following physical laws and terms also need to be
  included


                                   
- Newton's viscosity law               diffusive terms -
- Fourier's law of heat conduction     flux is a linear
                                       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
Transport equations                                   IFS
 Lagrangian formulation is simpler
                                            
                                           dx 
   - equation of the particle location        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
Transport equations                                     IFS
 - thermal energy conservation eq. for a particle

                        dT
                 mc p       QC  QL  QR
                        dt


                 convection        latent    thermal
                                    heat    radiation


 Conservation equations of Lagrangian model need to be
 solved for each representative particle




                                                              22
Transport equations                                                   IFS
Thermal radiation
                                     s


                        I(s)                       I(s+ds)
                               dA

                               Ie            Is




                  K a  K s I  K a I e  s  I s P'   d'
              dI                             K
              ds                             4 4 


      change of      absorption emission                 in-scattering
      radiation          and
       intensity    out-scattering
                                                                            23
Transport equations                                                              IFS
 Thermal radiation
Equations describing thermal radiation are much more
complicated
- spectral dependency of material properties
- angular (directional) dependence of the radiation transport

     dI   
                K a  K s I     K a I e  s    I ' P '   d'
                                                      K
                                                                 s    
        ds                                            4    4




  change of         absorption emission                          in-scattering
  radiation             and
   intensity       out-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
Averaging and simplification of
transport equations
                                                                  IFS
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
                       u                    
                     1                1                       1
             3   4                4                       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
Averaging and simplification of
transport equations
                                                                                     IFS
Transport equation variables can be decomposed onto
 a filtered (averaged) part and a residual (fluctuation)
        '           p  p  p'                          ~                    ~              ~
                                          vi  vi   vi*  vi  vi*        i  i  *   hi  h  h*


Filtered (averaged) transport equations
               ~
 t    j ( v j )  M                
                                  t   j  i vi  j  M j  i j 
                                      ~          ~~                                 turbulent mass
                                                                                     fluxes
                                                                                 sources and sinks
t  v j  i  vi v j    j p  2i Sij  g j  Fj  i ij 
      ~            ~~                                                            represent a separate
                                                                                 problem and require
                                                                                 additional models
                                 
t  h  i vi h  i iT  Q  i i 
     ~       ~~           ~
                                                                 - turbulent stresses
                                                                 - Reynolds stresses
                                 turbulent heat fluxes           - subgrid stresses
                                                                                                     29
Averaging and simplification of
transport equations
                                                                                        IFS
                                                                                     ~~
 Turbulent stresses                                              ji  v j vi   v j vi   v*j vi*

 Transport equation
     t   ji    k  vk  ji  
                           ~

          
      k p' v*j ik  p' vi*  jk   vi* v*j v*  v*jTik '  vi*T jk '
                                                k                          
     p' ( j vi*   i v*j )                                                     higher order
                 ~                ~                ~                ~
       ik  k v j    jk  k vi  F j vi  F j vi  Fi v j  Fi v j         product

                                                                                  turbulence
     Tik '  v  T jk '  v
                *
              k j
                                  *
                                k i                                               generation

     v*j  k  p ik  Tik   vi*  k  p  jk  T jk                        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
 Transport equation                                                                  ~~
                                                                     j  v j h   v j h    v*j h*

   t  j    k  vk  j  
                       ~

          
    k p' h*  jk  v* v*j h*  v*j qk '  h*Tkj '
                        k                                                higher order product
      
                ~
                             ~                
                                                ~           ~
   vk vj  k h  vk h  k v j  F j h  F j h  Q h  Q h               generation

   qk  k v*j  Tkj k h*                                                  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
Averaging and simplification of
transport equations
                                                   IFS
 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
                                                  IFS




(a)
                                    a)                   b)

      Temperature field comparison:
      a) steady-state RANS model, b) averaged LES model results
                                                              35
       Averaging and simplification of
       transport equations
                                                    IFS




(a)
                                 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
Zone models                                                   IFS
 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)

                 Qin (konv)                 Qout (konv+rad)
                              pg , Tg , mg , vg

                 mout     mf , H f                  min


                                                                    40
Zone models                                                           IFS
 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)   zgornja cona QU,out (konv+rad)
                                  pU,g , TU,g , mU,g , vU,g
            mU,out
            mL,out                spodnja cona                mL,in
                     mf , Hf      pL,g , TL,g , mL,g , vL,g
           Qin (konv)                                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
Field models                                                               IFS
Numerical grid and discretisation of transport equations
- simple example of discretisation
                      ui 1  ui                          ui 1  2ui  ui 1
          x u  lim                        x  xu  lim
                x 0     x                        x 0        x 2


- 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                                 ui 1  ui
      x u u   lim                         u  x u   lim ui 1 2
                x 0         x                            x 0            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
Field models                                                                      IFS
 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  x u x         x  x u 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
Field models                                        IFS
 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
Turbulence models                                         IFS
   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
Turbulence models                                                                   IFS
   Reynolds Averaged Navier-Stokes (RANS) models
    k-epsilon model
                                                                              
                       t k    j v j  k   P  G   j  (  t )  j k   
                                       ~
                                                                               
                                                                    k         


                            ~    C  P  C max(G ,0)     (  t )     C  
           t      j v j
                                                                                          2
                                                                
                                                               j            j 
                                                                                
                                                                       
                                      1        3                                     2
                                        k                                             k


     convection           production (source) term                           diffusion   destruction
                                                                                         (sink) term
    production             P  2t Sij  j vi   l vl  k  3t  l vl 
                                           ~ 2 ~                      ~
                                               3
    (source) term
                                             t
                                    G          g j  j  
                                            Prt
                                                                                                 58
Turbulence models                                                                   IFS
   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           2       3           k                t
                               0.09         1.44     1.92       1.0        1.0      1.3       0.9

 - from the calculated values of k in , eddy viscosity is
   defined as           2
                                       k
                           t  C ρ
                                        

 - from eddy viscosity, Reynolds stresses, turbulent heat and
   mass fluxes are obtained
            1                        2                                   t     ~             t ~
                                              ~
      ij    ll  ji  2t Sij  t ( l vl )  ji        j          jh   j          j
            3                        3                                   Prt                  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
Turbulence models                                                       IFS
   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

                              ~    P      (   2 C  k )     2 
        t   ij    l  vl ij
                                                                2
                                                
                                               l
                                                                        
                                                                     ij 
                                                              
                                       ij  ij              s       l            ij
                                                       3                3


                                   ~    C  P    (  t )     C  
                                                                               2
                t      l  vl       1
                                                     
                                                    l            l 
                                                                         2
                                               k                         k


      convection           generation       diffusion term                   destruction
                         (source) term                                       (sink) term
                                    pressure-velocity
                                     fluctuation term
       generation            Pij     ik   k u j     jk  k ui
                                                   ~                   ~
      (source) term
                                                                                           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
Turbulence models                                              IFS
   Large Eddy Simulation (LES) models
- required size of flow structures for LES modelling
                          1k
                                 32
              l  l EI   ~ 
                          6 

- 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 vi  Fi vi  Fi vi  




                                                                     65
Turbulence models                                                            IFS
   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 vl )  ji
                    3                     3
                                                                                     t ~
                                                                           j          jh
                                              and turbulent fluxes                   Prt

    the expression for turbulent viscosity can be written as
                                                                                           ~
     t  Cs  2S ji Sij  G               where the contribution              gi  i h
                  2                  1/ 2
                                                                                 G~       ~
                                                due to buoyancy is                  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
Turbulence models                                                                            IFS
              Comparison of turbulence models
                                          10                                                             10
                               k-e (Gr = 10 )                                                 k-e (Gr = 10 )
                               k-e N&B (Gr = 10 10 )                                          k-e N&B (Gr = 10 10 )
               8               RNG k-e (Gr = 10 10 )                                          RNG k-e (Gr = 10 10 )
               7               S S T (Gr = 10 )
                                               10
                                                                                              S S T (Gr = 10 )
                                                                                                              10
                                                10                                                             10
               6               S S G (Gr = 10 )                                               S S G (Gr = 10 )
                                               10                                                             10
               5               LES (Gr = 10 )                                                 LES (Gr = 10 )
                               Rous e e t al. (1952)                                          Rous e e t al. (1952)
               4               S habbir and Ge orge (1994)                                    S habbir and Ge orge (1994)
                                                                          10 -1
               3




                                                             2 1/3
1/3




                                                               b c (R /F0 )
 w c (R/F0 )




               2




                                                             5
               1                                                          10 -2




                    25    50     75            100                                25    50      75            100
                         z/R                                                           z/R


                         a)                                                            b)
Buoyant flow over a heat source: a) velocity, b) temperature*
                                                                                                                    68
             IFS


Combustion
 models




                   69
Combustion models                              IFS




                           Grinstein,
       Chen et al., 1988   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
Combustion models                                                                        IFS
 Modelling of composition requires solving n-1 transport
  equations for mixture components – mass or molar
  (volume) fractions

                                                                   ~ 
                     
                      ~           ~~      
                 t   j   i  vi  j   i    t
                                                 Sc Sc
                                                                       i  j   M j
                                                                              
                                                      t                     


 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
Combustion models                                                  IFS
 Reaction source term is defined as

      M j  Wj " j ' j R
                                or for multiple   M j  W j   " k , j  ' k , j Rk
                                reactions                      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
Combustion models                                       IFS
 Constant burning velocity sL
                    c
             sF       sL   speed of flame front propagation is
                    h
                            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            M j ~  f sL l


 - source/sink in energy transport equation           Q ~ M j hc


 - expressions for sL usually include additional models
                                                                          74
Combustion models                                        IFS
 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      sL ~  k
                                                         
  - Eddy break-up model reaction rate             R ~     ' 2
                                                              f
                                                         k

  - Eddy dissipation model reaction rate
                                           ~ o ~       ~
                                                        p        
                                     
                            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 
                  R  AT  exp  ~a    ' I  ~
                                                     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
Combustion models                                             IFS
 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 kg/s mixture
                              mešalni
   1-Z kg/s oxidiser                                      M
                              proces
       B

                                                   M  B
           Z  A  1  Z  B   M   or     Z
                                                    A  B

        Z is 1 in a fuel stream, 0 in an oxidiser stream
                                                                    81
Combustion models                                          IFS
 Flamelet model




 - conserved property often used in the mixture fraction
   definition    f  o i
 - for a simple chemical reaction
        1 kg 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
Combustion models                                                          IFS
  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  vi Z   i    t
                                                   Sc Sc
                                                               i Z 
                                                                    
                                                        t         


 - and a transport equation for variation of mixture fraction Z"

           ~2          ~ ~2                       ~' 2                          ~
       t   Z    i   ui Z    i   D  t
          
               ''
                       
                                ''
                                   
                                         
                                          
                                                 
                                                        i Z '   2
                                                       
                                                                     t
                                                                            ~2       '' 2
                                                                           i Z   C Z
                                     
                                         
                                                 Sct           
                                                                
                                                                      Sct             k


                                                                                               84
Combustion models                                           IFS
  Flamelet model
  - composition is calculated from preloaded libraries
                          1
                     j    j Z  PDF Z  dZ
                    ~
                          0


                  1
             j     j Z ,  PDF Z  PDF   dZ d
            ~
                  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
                       j  1  c  j ,svezi  c  j ,zgoreli
                      ~          ~ ~             ~~



                                                                 87
Combustion models                                                IFS
  Burning velocity model
                     Preheating               Oxidation layer
     T, 
            j
                Reaction layer O()
                                                           Temperature

                Oxidiser
                                                            Products
                 Fuel




                              Flame location                           x
        lF  0.1 ... 0.5 mm                             l  0.01 mm

                                                                           88
Combustion models                                                          IFS
  Burning velocity model
 - additional transport equation for the reaction progress
   variable
                       ~     v c       D   t
                 t  c          ~~         
                                                             ~
                                                             i c   M
                              i    i       i               
                                                    Sct    

                                                                           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
Thermal radiation                                                                 IFS
  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 a  K s I     K a I e  s    I ' P '   d'
                                                       K
                                                                  s    
         ds                                            4    4




   change of         absorption emission                          in-scattering
    intensity       and scattering
                                                                                        92
Thermal radiation                                                        IFS
  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                     E T   n T   E T  d [Wm-2 ]
                                                              2      4

                                                                         0
               the whole spectrum                                                    93
Thermal radiation                                               IFS
  Models
  - Rosseland (primitive model, for optical thick systems, additional
                   diffusion term in the energy transport equation)

               q  qd  qr  k d  k r  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
Thermal radiation                                                                          IFS
 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 , j   I  r , s  cos  j cos  d
                              rad

                                       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 (ANSYS-
  CFX, 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)
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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




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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




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