D r. Andrej Horvat Intelligent Fluid Solutions Ltd. Ljubljana , Slovenia 17 January, 200 8

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D r. Andrej Horvat Intelligent Fluid Solutions Ltd. Ljubljana , Slovenia 17 January, 200 8

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ComputationalModelsfor Prediction of Fire Behaviour

Dr. Andrej Horvat

Intelligent Fluid Solutions Ltd.

Ljubljana, Slovenia

17 January, 2008

Contact information

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

Personal information

- 1995, Dipl. -Ing. Mech. Eng. (Process Tech.)
Universityof Maribor

- 1998, M.Sc. Nuclear Eng.
Universityof Ljubljana

- 2001, Ph.D. Nuclear Eng.
Universityof Ljubljana

- 2002, M.Sc. Mech. Eng. (Fluid Mechanics & Heat Transfer)
University of California, Los Angeles

Personal information

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)

Personal information

As well as CFD, experiences also in:

- Experimental methods
- QA procedures
- Standardisation and technical regulations
- Commercialisation of technical expertise and software products

Contents

- 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

Contents

- 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

Some basic thoughts …

- 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 costand 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)

Overview of fluid dynamics

transport equations

- The continuum assumption

Transport equations

-A control volume has to contain a largenumber of particles(atoms or molecules): Knudsen number << 1.0

- At equal distribution of particles (atoms or molecules) flow quantitiesremain unchangeddespite of changes in location and size of a control volume

- Eulerian andLagrangian description

Transport equations

- Eulerian description– transport equations formass, momentumand energy are written fora (stationary) control volume
- Lagrangian description– transport equations for mass, momentum and energy are written for a moving material particle

- Transport of mass and composition
- Transport of momentum
- Transport of energy

- 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
- Lagrangianformulationis basis for modelling of particle dynamics: bubbles, droplets (sprinklers), solid particles (dust) etc.

Transport equations

Transport equations

Droplets trajectoriesfrom sprinklers (left), gas temperature field during fire suppression (right)

- Eulerian formulation of mass transport equation
- integral form differential (weak) form
- The equation also appears in the following forms
- non-conservative form

Transport equations

flux difference

(convection)

change of massin a control vol.

~ 0

in incompressible fluid flow

- Eulerian formulation of mass fraction transport equation (general form)

change of mass of a component in a control vol.

flux difference

(convection)

diffusive mass flow

- Eulerian formulation of momentum equation

Transport equations

viscous

force

(diffusion)

change of momentum in a control vol.

volumetric force

pressure force

flux difference

(convection)

- Eulerian formulation of energy transport equation

Transport equations

change of internal energy in a control vol.

flux difference

(convection)

deformation work

diffusive heat flow

- The following physical laws and terms also need to be included
- Newton's viscosity law

- Fourier'slaw of heat conduction
- Fick's law of mass transfer
- Sources and sinksdue to thermal radiation, chemical reactions etc.

diffusive terms -

flux is a linear function of a gradient

- Transport of mass and composition
- Transport of momentum
- Transport of energy

- Lagrangianformulationis simpler
- equation of the particle location

- mass conservation eq. for a particle

- momentum conservation eq. for a particle

Transport equations

drag

lift

volumetric forces

- thermal energy conservation eq. for a particle

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

Transport equations

convection

latent

heat

thermal radiation

- Thermal radiation

Transport equations

s

I(s) I(s+ds)

dA

Ie Is

change of radiation

intensity

absorption and

out-scattering

emission

in-scattering

- Thermal radiation
Equations describing thermal radiation are much more complicated

- spectral dependency of material properties

- angular (directional) dependence of the radiation transport

Transport equations

in-scattering

change of radiation

intensity

absorption and

out-scattering

emission

Averaging and simplification

of transport equations

- The presented set of transport equations is analytically unsolvable for the majority of cases
- Success of a numerical solving procedure is based ondensity of the numerical grid, and in transient cases, also on the size of the integration time-step
- Averagingandsimplification 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

Averaging and simplification of transport equations

Averaging and simplification of transport equations

- Averaging and filtering
The largest flow structures can occupy the whole flow field, whereas the smallest vortices have the size of Kolmogorov scale

,vi , p, h

w

,

Averaging and simplification of transport equations

- Kolmogorov scaleis (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

- - time interval k-epsilon model, SST model,
- Reynolds stress models

Averaging and simplification of transport equations

- Transport equation variables can be decomposed onto a filtered (averaged) partanda residual (fluctuation)
- Filtered (averaged) transport equations

turbulent mass fluxes

sources and sinks represent a separate problem and require additional models

- turbulent stresses

- Reynolds stresses

- subgrid stresses

turbulent heat fluxes

Averaging and simplification of transport equations

- Turbulent stresses
- Transport equation
- the equation is not solvable due to the higher order product

- all turbulence models include at least some of the termsof this equation (at least the generation and the dissipation term)

higher order

product

turbulence generation

turbulence dissipation

Averaging and simplification of transport equations

- Turbulent heat and mass fluxes
- Transport equation
- the equation is not solvable due to the higher order product

- most of the turbulence models do not take into account the equation

higher order product

generation

dissipation

Averaging and simplification of transport equations

- Turbulent heat fluxesdue 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

Averaging and simplification of transport equations

Buoyancy induced flow over a heat source (Gr=10e10); inert model of a fire

Averaging and simplification of transport equations

(a)

LES model; instantaneous temperature field

(b)

Averaging and simplification of transport equations

a) b)

Temperature field comparison:

a) steady-state RANS model, b) averaged LES model results

(a)

(b)

Averaging and simplification of transport equations

a) b)

Comparison of instantaneousmass fraction in a gravity current : a) transient RANS model, b) LES model

(a)

(b)

Averaging and simplification of transport equations

- Additional simplifications
- flow can be modelled as a steady-state case the solution is a result of force, energy and mass flow balancetaking 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 transferis modelled as asimple sinkof thermal energy FDStakes 35% of thermal energy

- control volumescan be solarge that continuity of flow properties is not preserved zone models

Zone models

- 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 modelstake 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

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

Zone models

Qw (konv+rad)

Qout (konv+rad)

Qin (konv)

pg , Tg , mg , vg

mout

min

mf , Hf

- One-zone and two-zone models
- two-zone modelscan be used for evaluation of a localised fire before flashover

- a room is separated onto different zone, most often onto an upperandlower zone, afireandbuoyant flowof gases above the fire

- conditions are uniform and constant in each zone

Zone models

QU,out (konv+rad)

Qw (konv+rad)

zgornja cona

pU,g , TU,g , mU,g , vU,g

mU,out

spodnja cona

pL,g , TL,g , mL,g , vL,g

mL,out

mL,in

mf , Hf

QL,out (konv+rad)

Qin (konv)

- 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

Zone models

Field models

- Numerical grid and discretisation of transport equations
- analytical solutionsof transport equations are known only for few very simple cases

- for most real world cases,one needs to use numerical methodsand 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 calleddiscretisation.The following methods areused: finite difference method, finite volume method, finite element methodandboundary element method (and different hybrid methods)

Field models

- Numerical grid and discretisation of transport equations
- simple example of discretisation

- many different discretisation schemes exist; they can be divided ontoconservativeandnon-conservative schemes (linked to the discrete form of the convection term)

Field models

- Numerical grid and discretisation of transport equations
- non-conservative schemesrepresent a linear form and therefore they are more stable and numericallybetter manageable

- non-conservative schemesdo not conserve transported quantities, which can lead to time-shift of a numerical solution

Field models

time-shift due to a non-conservative scheme

- Numerical grid and discretisation of transport equations
- qualityof numerical discretisation is defined with discrepancy between a numerical approximation and an analytical solution

- error is closely link to the orderof discretisation

Field models

1st order truncation (~x)

- 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: dissipationanddispersion

Field models

- 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 alsohigher order schemes (ENO, WENO etc.) but their use is limited on structured numerical meshes

Field models

i-1, j+1

i, j+1

i+1, j+1

i, j

i-1, j

i+1, j

i, j-1

i+1, j-1

i-1, j-1

- Numerical grid and discretisation of transport equations
- connection matrix and location of neighbouring grid nodes defines two types of numerical grid: structured and unstructurednumerical grid

structured grid unstructuredgrid

Field models

k+2

k+1

k+6

k

k+9

k+3

k+5

k+7

k+4

- Numerical grid and discretisation of transport equations
- unstructured gridsraise 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

Field models

- Numerical grid and discretisation of transport equations

Field models

- 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

Turbulence

models

Turbulence models

laminar flow

transitional flow

turbulentflow

van Dyke, 1965

- 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 modelsto use less dense (smaller) numericalgrid(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)

Turbulence models

- 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 atime interval Reynolds Averaged Navier-Stokes (RANS) models: k-epsilon model, SST model, Reynolds Stress modelsetc
- ForRANS models, size of the averaging time interval is not known or given (statistical average of experimental data)
- ForLES models, size of the filter or the spatial averaging interval isa basic input parameter (in most case it is equal to grid spacing)

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

- for dissipation of turbulent fluctuations

or

- for frequency of turbulent fluctuations

- Reynolds Averaged Navier-Stokes (RANS) models
k-epsilon model

production

(source) term

production (source) term

destruction (sink) term

convection

diffusion

- 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

- from the calculated values of k in , eddy viscosity is defined as

- from eddy viscosity, Reynolds stresses, turbulent heat and mass fluxes are obtained

- Reynolds Averaged Navier-Stokes (RANS) models
- transport equation for k is derived directly from the transport equations for Reynolds stresses

- 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

- Reynolds Averaged Navier-Stokes (RANS) models
For Reynolds Stress(RS) model, 7 additional transport equations need to be solved:

- for6 componentsof Reynolds stress tensor

- for dissipation of turbulent fluctuations

or

- for frequency of turbulent fluctuations

- Reynolds Averaged Navier-Stokes (RANS) models
Reynolds stress model

generation

(source) term

convection

generation (source) term

diffusion term

destruction (sink) term

pressure-velocity fluctuation term

- 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

- describesanisotropic 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)

- 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).

Turbulence models

- Large Eddy Simulation (LES) models
- required size of flow structures for LES modelling

- 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

Turbulence models

- Large Eddy Simulation (LES) models
- eddy (turbulent) viscosity is defined as

where

- using the definition of turbulent (subgrid) stresses

and turbulent fluxes

the expression for turbulent viscosity can be written as

where the contribution

due to buoyancy is

Turbulence models

grid spacing

- 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

Turbulence models

- Comparison of turbulence models

Turbulence models

a) b)

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

Combustion

models

Grinstein,

Kailasanath, 1992

Chen et al., 1988

- 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

- Modelling of composition requires solving n-1 transport equations for mixture components – mass or molar (volume) fractions
- Chemical balance equationcan be written as
or

- Reaction source termis defined as
or for multiple

reactions

whereR orRkis a reaction rate

- Reaction rateis 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

- Constant burning velocitysL
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/sinkin mass fraction transport equation

- source/sink in energy transport equation

- expressions forsLusually include additional 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

- Eddy break-up model reaction rate

- Eddy dissipation model reaction rate

- Eddy break-up model and Eddy dissipation model
- typical values of model coefficientsCA= 4 in CB= 0.5

- the model can be used for simple reactions (one-andtwo-stepcombustion)

- 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 scaleetc.

- Backdraft simulation

- Finite rate chemistry model
- - it is applicable when a chemical reaction rate is slow or comparable with turbulent mixing
- reaction kinetics must be known

- for each additional component, the component molar concentration Ineeds to be multiplied with the product

- for each additional reaction the same expression is added

- 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

- 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

- Flamelet model
- - the model is only applicable for two-feed systems (fuelandoxidiser)
- - it is based on definition ofa mixture fraction
- or

Zkg/s fuel

A

mešalni

proces

1 kg/s mixture

1-Zkg/s oxidiser

M

B

Zis 1 in a fuel stream, 0 in an oxidiser stream

- Flamelet model

- conserved property often used in the mixture fraction definition

- - for a simple chemical reaction
- the stoihiometricratio is

- Flamelet model
- Shvab-Zel'dovich variable

- the conditions in vicinity of flamelets are described with the respect to Z; Z=Zstis a surface with the stoihiometricconditions

- transport equationsare rewritten withZdependencies;

conditions areone-dimensionalξ(Z) , T(Z) etc.

- for limited cases, the simplified equations can be solve analytically

- Flamelet model
- numerical implementations of the model differ significantly

- for turbulent flow, we need to solve an additional transport equation for mixture fraction Z

- and atransport equationforvariation of mixture fraction Z"

- Flamelet model
- composition is calculated from preloaded libraries

- is the scalar dissipation rate

- it increases with stresses(stretching of a flame front)

anddecreases with diffusion.

- at critical value of flame extinguishes

these PDFs are tabulated for differentfuel, oxidiser, pressure and temperature

- Flamelet model

Ferreira, Schlatter, 1995

- 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)orTurbulent 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

- Burning velocity model

Preheating

Oxidation layer

T,

j

Reaction layer O()

Temperature

Oxidiser

Products

Fuel

x

Flame location

- Burning velocity model
- additional transport equation for the reaction progress variable

- source/sinkdue to combustion is defined as

- better results by solving the transport equation for weighted reaction progress variable

source term

modelling of the turbulent burning velocity

- 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

Thermal radiation

models

- 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

in-scattering

change of intensity

absorption and scattering

emission

- 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
- frequency

c - speed of light

n - refraction index

h - Planck's constant

kB - Boltzmann's constant

integration over

the whole spectrum

- Models
- Rosseland (primitive model, for optical thick systems, additional

diffusion term in the energy transport equation)

- P1 (strongly simplified radiation transport equation - solution of an

additional Laplace equation is required)

- Discrete Transfer (modern deterministic model,

assumes isotropic scattering, reasonably

homogeneous properties)

- Monte Carlo(modern statistical model, computationally

demanding)

- 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
- source/sink in the energy transport equation

- 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 Kais 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"

- 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

- Flashover simulation

Conclusions

- The seminar gave a short (but demanding) overview of fluid mechanics and heat transfer theory that is relevant for fire simulations
- All currentcommercial 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)

- 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

Acknowledgement

- I would like to thank to ourhostsand especially to Aleš Jug
- I would like to thank to ANSYS Europe Ltd.(UK) who permitted access to some of the graphical material