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**Computational Models for Prediction of Fire Behaviour**

Dr. Andrej Horvat Intelligent Fluid Solutions Ltd. Ljubljana, Slovenia 17 January, 2008

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Contact information Andrej Horvat Intelligent Fluid Solution Ltd. 127 Crookston Road, London, SE9 1YF, United Kingdom Tel./Fax: +44 (0) Mobile: +44 (0) Web:

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**Personal information 1995, Dipl. -Ing. Mech. Eng. (Process Tech.)**

University of Maribor 1998, M.Sc. Nuclear Eng. University of Ljubljana 2001, Ph.D. Nuclear Eng. 2002, M.Sc. Mech. Eng. (Fluid Mechanics & Heat Transfer) University of California, Los Angeles

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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)

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**Personal information As well as CFD, experiences also in:**

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

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**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

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**Contents Field models Conclusions Examples**

- 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

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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 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)

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**Overview of fluid dynamics**

transport equations

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**Transport equations 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

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**Transport equations 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

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**Transport equations Transport of mass and composition**

Transport of momentum Transport of energy

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Transport equations 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.

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Transport equations Droplets trajectories from sprinklers (left), gas temperature field during fire suppression (right)

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**Transport equations Eulerian formulation of mass transport equation**

integral form differential (weak) form The equation also appears in the following forms non-conservative form flux difference (convection) change of mass in a control vol. ~ 0 in incompressible fluid flow

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Transport equations 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

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**Transport equations Eulerian formulation of momentum equation**

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

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**Transport equations Eulerian formulation of energy transport equation**

change of internal energy in a control vol. flux difference (convection) deformation work diffusive heat flow

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Transport equations The following physical laws and terms also need to be included - Newton's viscosity law Fourier's law of heat conduction Fick's law of mass transfer Sources and sinks due to thermal radiation, chemical reactions etc. diffusive terms - flux is a linear function of a gradient

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**Transport equations Transport of mass and composition**

Transport of momentum Transport of energy

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**Transport equations Lagrangian formulation is simpler**

- equation of the particle location - mass conservation eq. for a particle - momentum conservation eq. for a particle drag lift volumetric forces

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**Transport equations - thermal energy conservation eq. for a particle**

Conservation equations of Lagrangian model need to be solved for each representative particle convection latent heat thermal radiation

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**Transport equations Thermal radiation change of radiation intensity**

I(s) I(s+ds) dA Ie Is change of radiation intensity absorption and out-scattering emission in-scattering

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**Transport equations Thermal radiation**

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

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**Averaging and simplification of transport equations**

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**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 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

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**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 ,

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**Averaging and simplification of transport equations**

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

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**Averaging and simplification of transport equations**

Transport equation variables can be decomposed onto a filtered (averaged) part and a 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

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**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 terms of this equation (at least the generation and the dissipation term) higher order product turbulence generation turbulence dissipation

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**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

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**Averaging and simplification of transport equations**

Turbulent heat fluxes due to thermal radiation - little is known and published on the subject - majority of models do not include this contribution - radiation heat flow due to turbulence

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**Averaging and simplification of transport equations**

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

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**Averaging and simplification of transport equations**

LES model; instantaneous temperature field (b)

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**Averaging and simplification of transport equations**

a) b) Temperature field comparison: a) steady-state RANS model, b) averaged LES model results (b)

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**Averaging and simplification of transport equations**

a) b) Comparison of instantaneous mass fraction in a gravity current : a) transient RANS model, b) LES model (b)

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**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 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

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Zone models

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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 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

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**Zone models One-zone and two-zone models**

- one-zone models can be used only for assessment of a fully developed fire after flashover - in such conditions, a valid approximation is that the gas temperature, density, internal energy and pressure are (more or less) constant across the room Qw (konv+rad) Qout (konv+rad) Qin (konv) pg , Tg , mg , vg mout min mf , Hf

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**Zone models One-zone and two-zone models**

- two-zone models can be used for evaluation of a localised fire before flashover - a room is separated onto different zone, most often onto an upper and lower zone, a fire and buoyant flow of gases above the fire - conditions are uniform and constant in each zone 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)

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**Zone models 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

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Field models

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**Field models 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)

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**Field models Numerical grid and discretisation of transport equations**

- simple example of discretisation - many different discretisation schemes exist; they can be divided onto conservative and non-conservative schemes (linked to the discrete form of the convection term)

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**Field models 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

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**Field models Numerical grid and discretisation of transport equations**

- quality of numerical discretisation is defined with discrepancy between a numerical approximation and an analytical solution - error is closely link to the order of discretisation 1st order truncation (~x)

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**Field models 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

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**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 also higher order schemes (ENO, WENO etc.) but their use is limited on structured numerical meshes

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**Field models Numerical grid and discretisation of transport equations**

- connection matrix and location of neighbouring grid nodes defines two types of numerical grid: structured and unstructured numerical grid structured grid unstructured grid i, j i+1, j i-1, j i, j+1 i-1, j+1 i-1, j-1 i, j-1 i+1, j-1 i+1, j+1 k+2 k+1 k+6 k+9 k k+3 k+7 k+5 k+4

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**Field models 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

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**Field models 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

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Turbulence models

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**Turbulence models transitional flow turbulent flow laminar flow**

van Dyke, 1965

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Turbulence models 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)

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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 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)

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**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

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**Turbulence models Reynolds Averaged Navier-Stokes (RANS) models**

k-epsilon model production (source) term convection production (source) term diffusion destruction (sink) term

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**Turbulence models Reynolds Averaged Navier-Stokes (RANS) models**

- model parameters are usually defined from experimental data e.g. dissipation of grid generated turbulence or flow in a channel - from the calculated values of k in , eddy viscosity is defined as - from eddy viscosity, Reynolds stresses, turbulent heat and mass fluxes are obtained C C1 C2 C3 k Prt 0.09 1.44 1.92 1.0 1.3 0.9

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**Turbulence models 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

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**Turbulence models Reynolds Averaged Navier-Stokes (RANS) models**

For Reynolds Stress (RS) model, 7 additional transport equations need to be solved: - for 6 components of Reynolds stress tensor - for dissipation of turbulent fluctuations or - for frequency of turbulent fluctuations

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**generation (source) term pressure-velocity fluctuation term**

Turbulence models 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

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**Turbulence models 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 - 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)

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**Turbulence models 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).

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**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

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**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 grid spacing

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**Turbulence models 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

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**Turbulence models Comparison of turbulence models a) b)**

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

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Combustion models

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Combustion models Grinstein, Kailasanath, 1992 Chen et al., 1988

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**Combustion models 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

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

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**Combustion models Reaction source term is defined as or for multiple**

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

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**Combustion models Constant burning velocity sL**

speed of flame front propagation is larger due to expansion - values are experimentally determined for ideal conditions - limits due reaction kinetics and fluid mechanics are not taken into account - source/sink in mass fraction transport equation - source/sink in energy transport equation - expressions for sL usually include additional models

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**Combustion models Eddy break-up model and Eddy dissipation model**

- is a well established model - based on the assumption that the reaction is much faster than the transport processes in flow - reaction rate depends on mixing rate of reactants in turbulent flow - Eddy break-up model reaction rate - Eddy dissipation model reaction rate

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**Combustion models 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.

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Combustion models Backdraft simulation

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**Combustion models 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 I needs to be multiplied with the product - for each additional reaction the same expression is added

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**Combustion models 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

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**Combustion models 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

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**Combustion models Flamelet model**

- the model is only applicable for two-feed systems (fuel and oxidiser) - it is based on definition of a mixture fraction or Z kg/s fuel A mešalni proces 1 kg/s mixture 1-Z kg/s oxidiser M B Z is 1 in a fuel stream, 0 in an oxidiser stream

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**Combustion models Flamelet model**

- conserved property often used in the mixture fraction definition - for a simple chemical reaction the stoihiometric ratio is

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**Combustion models Flamelet model - Shvab-Zel'dovich variable**

- 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

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**Combustion models Flamelet model**

- numerical implementations of the model differ significantly - for turbulent flow, we need to solve an additional transport equation for mixture fraction Z - and a transport equation for variation of mixture fraction Z"

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**Combustion models Flamelet model**

- composition is calculated from preloaded libraries - is the scalar dissipation rate - it increases with stresses (stretching of a flame front) and decreases with diffusion. - at critical value of flame extinguishes these PDFs are tabulated for different fuel, oxidiser, pressure and temperature

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Combustion models Flamelet model Ferreira, Schlatter, 1995

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**Combustion models 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

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**Combustion models Burning velocity model T, Temperature Oxidiser**

Preheating Oxidation layer T, j Reaction layer O() Temperature Oxidiser Products Fuel x Flame location

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**Combustion models Burning velocity model**

- additional transport equation for the reaction progress variable - source/sink due 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

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**Combustion models 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

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Thermal radiation models

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**absorption and scattering**

Thermal radiation It is a very important heat transfer mechanism during fire In fire simulations, thermal radiation should not be neglected The simplest approach is to reduce the heat release rate of a fire (35% reduction in FDS) Modelling of thermal radiation - solving transport equation for radiation intensity change of intensity absorption and scattering emission in-scattering

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Thermal radiation Radiation intensity is used for definition of a source/sink in the energy transport equation and radiation wall heat fluxes Energy spectrum of blackbody radiation - frequency c - speed of light n - refraction index h - Planck's constant kB - Boltzmann's constant integration over the whole spectrum

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**Thermal radiation 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)

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**Thermal radiation Discrete Transfer - modern deterministic model**

- assumes isotropic scattering, homogeneous gas properties - each wall cell works as a radiating surface that emits rays through the surrounding space (separated onto multiple solid angles) - radiation intensity is integrated along each ray between the walls of the simulation domain - source/sink in the energy transport equation

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**Thermal radiation 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"

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**Thermal radiation 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

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Thermal radiation Flashover simulation

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Conclusions

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Conclusions 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 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 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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