CHAPTER 11 The implementation and application of a ﬁ re ......These include sprinklers, water mist...

CHAPTER 11

The implementation and application of a fi re CFD model

J. Trelles & J.E. FloydHughes Associates, Inc., Baltimore, MD, USA.

Abstract

The previous chapters have presented the details of various fi re transport phenomena. The present chapter demonstrates how to put these theoretical and experimental building blocks together into a viable computational fl uid dynamics (CFD) model and then how to effectively use the model. The emphasis is on the Fire Dynamics Simulator (FDS), a leading model in the fi eld of fi re CFD which also has the highest usership. First, the aerodynamic fundamentals such as the low Mach number equations for expandable fl ow and the manipulation of the equations for the purposes of discretization are presented. Two different approaches to turbulence are discussed: direct and large eddy simulations (LESs). The emphasis then shifts to heat sources and heat sinks for the CFD models. These are what really make this a fi re CFD model. Two methods are presented for distributed heat input from unconfi ned combustion sources: solution of transport equations with Arrhenius terms for direct simulations and a mixture fraction model for LESs. The radiation model in FDS allows for heat transfer from fl ames and hot layers without doubling the computa-tional overhead. Heat extraction methods are implemented for a variety of fi re protection systems. These include sprinklers, water mist systems, and smoke exhaust. The last topic encompasses the effective application of what is really a complicated model. The fi rst step is appropriate modelling of the scenario in question. The collection of input data is always a challenge in fi re CFD because many items, such as the composition of a fuel, are unknown. Good results are often obtained with a suffi ciently simple model that captures all the important physical contributions. Checking the work against known results from the literature is important to ensure reasonable predictions. Examples are given of how a good model can be changed into a bad model by the injudicious choice of one input variable.

1 Introduction

The preceding chapters have detailed the physical and chemical fundamentals of fi re phenom-ena. In this chapter, it is shown how these concepts are integrated into a comprehensive fi re

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408 Transport Phenomena in Fires

dynamics simulation. Recall that a computational fl uid dynamics (CFD) model is based on the Navier−Stokes equations,

( ),v

t

rr

∂= ∇⋅

∂

(1)

2 ,Dv

p v gDt

r m r= −∇ + ∇ +

(2)

( ) .

Duq q v v v g

Dtr r t r= −∇⋅ + − ∇⋅ + ∇⋅ ⋅ + ⋅′′ ′′′

(3)

Here t is the time, ∇ is the gradient with respect to the space vector x, r is the density, v

is the

velocity vector, p is the pressure, µ is the viscosity coeffi cient, g is the acceleration of gravity, u is the internal energy, q′′ is the heat fl ux vector, q′′′ is the volumetric heat generation term, and __

__ t is the stress tensor.

When using these equations to model a fi re, a striking observation is made after considering the phenomena to be addressed. In a fi re, the volumetric energy source term, q′′′ , in these equations is combustion. A fi re is a complex series of hundreds of chemical reactions occurring on 10−6 m or less length scales and 10−6 s or less timescales. Density changes and velocity can be driven by either local temperature changes or by pressure waves. Therefore, the velocity term can range in magnitude from buoyancy driven fl ows, 10 m/s, to sonic fl ows, 103 m/s. If the problem of interest is, for example, a wildland fi re, then the overall problem length can be 104 m, and the time can be 106 s, 1 week. To create a computer model of fi re, therefore, means effi ciently solving the Navier−Stokes equations over potentially 12 or more orders of magnitude in time and 10 or more orders of magnitude in length. It is clear that simplifi cations must be made to make a solution computa-tionally feasible.

Even when simplifi ed, a number of complex issues remain. Sources and sinks, such as volu-metric heat addition, momentum impact terms induced by the transport of condensed phases, and various boundary conditions (BCs), serve to convert a basic aerodynamic solver into fi re physics platform. A number of additional physical models are required to populate the variables in the Navier−Stokes equations. The heat release rate (HRR) term must be determined. Heat transfer to surfaces must be determined. Additionally, a fi re protection engineer will wish to include the effects of suppression or the response of installed detection systems. Each of these additional models can be made very complex or very simple.

To create a fi re model, therefore, is an exercise in determining the appropriate level of com-plexity in the various physical models to reach the desired solution accuracy while consuming a reasonable quantity of computational resources (CPUs and/or time). For concreteness, the emphasis will be on the Fire Dynamics Simulator (FDS), a large eddy simulation (LES) devel-oped and maintained by the National Institute of Standards and Technology (NIST) [1]. FDS is a multidimensional, multiphysics CFD simulation. It can handle isothermal or thermally variable fl ows. It has options for the direct simulation of turbulence or for LESs. It can accommodate axisymmetric cylindrical, two-dimensional and three-dimensional Cartesian coordinates. The approach can be summarized by three maxims:

The required computing power should be affordable to a typical fi re protection fi rm ($1,000s 1. rather than $10,000s).The required computing time should lie within the length of time available to a typical fi re 2. protection project (days or weeks rather than months).



The Implementation and Application of a Fire CFD Model 409

Computational time for a submodel should be proportional to the importance of that sub-3. model in the fi nal solution (this has been referred to as Baum’s rule).

This chapter will also address the practice of CFD modelling. Obtaining a good series of simulations is a challenging objective in general that, for fi re problems, is exacerbated by the uncertainty associated with characterizing such important items as fuels and HRRs. The tech-nique of good modelling is detailed along with the methods of analysis that can be employed to certify the fi delity of a solution set. Since the readily available FDS theory manual [1] and the user’s manual [2] form an excellent introduction to the FDS, the authors of this chapter have made a conscious effort to avoid topics covered therein, instead concentrating, as much as pos-sible, on relevant material not developed elsewhere.

2 Turbulence modelling

Fires require air to burn. Air and fuel are mixed via turbulent eddies and diffusion in the fl ow fi eld. Too much mixing will result in short fl ame heights and cooler plumes. Too little mixing and the opposite will happen. Turbulence is formed by shear within the fl ow resulting from either wall friction or density gradients. Turbulent structures within a fl ame can be less than 1 mm, whereas those within a large plume can be greater than 1 m. Thus, if the primary concern is the fl ame then very small turbulent structures must somehow be resolved. If the fi re plume or other larger-scale fi re phenomena are the main interest then only the larger eddies need be resolved. Within FDS, both options are available to the user. Direct numerical simulation (DNS) can be used when the grid resolution is capable of resolving the smallest eddies. LES can be used for larger grids.

When performing a DNS computation, no submodels are used for the purpose of creating turbulence in the computed fl ow fi eld. Provided suffi cient grid resolution is used and appropriate BCs are specifi ed, the proper fl ow fi eld will be computed. While using DNS can avoid the uncer-tainties of a turbulence submodel containing empirical constants, it imposes a signifi cant cost on the user. When performing a CFD computation, the solution for the next time step uses the infor-mation from the prior time step. If that information traverses multiple grid cells in a time step, velocity × time step > grid size, then the solution can become unstable. For a DNS fi re simula-tion with the grid resolution of the order of a millimetre and fl ow speeds of the order of 10 m/s, the time step will be < 0.1 ms. Therefore, if one could update a grid cell per 0.1 µs, then using DNS on a single CPU to compute a fi re in a 1 m3 cube for one minute would take almost 2 years. DNS, therefore, is not practical for fi re protection applications. Nonetheless, a few exemplars exist such as the laboratory bench-scale studies published in [3] and [4] and the slot-burner examples presented below.

The other option available to a user of FDS is LES. In an LES computation the grid is selected to resolve the dominant eddy structures in the fl ow, which for a fi re are driven by the plume, and all smaller eddy structures are handled by a subgrid scale (SGS) model. Multiple SGS models exist, and FDS uses a fairly simple model: the Smagorisnky SGS model. In the Smagorisnky model, the effect of SGS turbulence is accounted for by computing an effective local viscosity. This effective viscosity is given by

2 2LES s

2( ) 2(def ) (def ) ( ) .

3C u u um r = ∆ ⋅ − ∇⋅

(4)

The parameter ∆ is the mean length of the local grid cell, def is the symmetric gradient operator, and the parameter Cs is an empirical constant. FDS uses a constant Cs. Experimentally Cs will




vary depending upon where in the fl ow one is evaluating the turbulent viscosity. However, for the large-scale fi res typically modelled with FDS the assumption of a constant Cs has yielded good results in the validation studies [37] that have been performed to date. One question that often arises is how small or large can the grid, and hence ∆, be and still have an accurate solution. For large-scale fi re simulations it is recommended [1] to resolve the characteristic size of the fi re, D*, with as many cells as possible,

2

5

a a p

.Q

D*T c gr

=

(5)

This, however, is a rule of thumb and is not a substitute for a grid study to verify a converged solution.

3 Solution speed and stability

CFD fi re models, while capable of producing highly resolved output and supporting realistic ren-derings of mass and energy fl ows, are also very slow when compared to other methods of compu-tation. This speed can often eliminate the use of CFD as a tool for a particular commercial project or greatly limit the range of scenarios to be examined. Therefore, assumptions and approximations that can reduce the computational time are in order.

In many CFD codes, the computational time is driven by the pressure solver. For example, in FDS, the perturbation pressure is determined by solving a Poisson equation with an involved right-hand side. Since FDS uses a two stage Runge−Kutta method for each time step update, the Poisson solver is called twice at each time step. The pressure fi eld is tightly coupled with the velocity fi eld. The infl uence of the pressure fi eld is felt in the form of internal waves and acoustic waves. Internal wave speeds are of the same order as typical non-premixed fi re-related fl ow speeds. These internal waves can be associated with the pressure potential that is driving the low-Mach number fl ow. However, the speed of sound of about 350 m/s is substantially greater than typical compartment fi re fl ows. In spite of this velocity dominance, for low-Mach number fl ows the speed of sound is related to acoustic waves which have little impact on the fl ow and the sur-roundings (unlike shock waves which would dominate the fl ow). In other words, for the compart-ment fi re fl ow regime, the acoustic waves are essentially naturally decoupled from the ensuing velocity fi eld. On the other hand, FDS’s explicit time-stepping implies that, because of the Cou-rant−Friedrichs−Levy (CFL) condition, the time step is inversely proportional to the fastest speed. FDS addresses this dilemma by fi rst using a subset of the Navier−Stokes equations which eliminate acoustic waves. The expandable gas equations [1, 6] can formally be derived using a Mach number expansion. Equations (1)−(3) are non-dimensionalized according to the parame-ters outlined in [7]. The dependent variables are expanded in powers of Mach number, M [7, 8]. By making the appropriate substitutions and taking the limit M→0, it is established that the low-est order terms for the pressure are only functions of time while the next order term is a function of x and t. Hence the following decomposition of the pressure results,

0( , ) ( ) ( , ).p x t P t p x t= + (6)

Note that p( x

,t) is often decomposed into –gzêz + p(x

,t) in order to facilitate outdoor simulations where the background weather dominates the fl ow. Equation (6) is responsible for fi ltering out sound waves in the resulting system of equations. It is also a generalization of the commonly




used gauge pressure decomposition. In fact, for open domains, P0(t) is constant and the standard gauge pressure decomposition is obtained.

Rehm and Baum [6] worked with inviscid equations and used different non-dimensionaliza-tion and expansion parameters. Reference [6] contains a dispersion equation obtained from the expandable equations’ limit. It is shown to be consistent with the dispersion relation for a fl uid with internal waves but no acoustic waves. Hence the two manifestations of the pressure fi eld were not only decoupled but the undesirable acoustic waves were eliminated. The low Mach number family of equations have also been investigated for mathematical well-posedness and short-time existence of solutions [9−11].

It has already been mentioned that eliminating acoustic waves increases the time steps allowed by the CFL condition. In order to get an idea of just how pervasive acoustic waves are, consider the example of a fi re within a single compartment with a slightly open window and an initially closed door. The act of simply opening the door sends out acoustic waves. The fi re crackle is also a source of constant sound waves. Typically, fi re investigators are not interested in tracking the dynamics and fronts of these waves. The disadvantage to the expandable gas equations is that strong defl agrations, detonations, and high speed jets cannot be modelled. The rule of thumb for applicability [1] is to stay within the incompressible regime for which M < 0.3 [12].

FDS’s procedure for updating the pressure is closer to that used in projection methods rather than that used in SIMPLE algorithms. The resulting Poisson equation does not have constant coeffi cients so an approximation is made that allows an effi cient, fast Fourier transform (FFT)-based solver to be used. The choice of Poisson solver has had a large impact on the design of FDS. It has resulted in an orthogonal coordinate system that can only have two axes with variable gridding. In order to accom-modate mixed BCs, the pressure solver is implemented with deferred corrections based on the previ-ous time step’s velocity fi eld. When more fl exibility was required in the types of domains that could be modelled, the desire to maintain the same Poisson solver resulted in multiblocked domains and further corrections in order to match the pressures across subdomain interfaces.

4 Accounting for energy

Modelling a fi re entails the incorporation of various energy sources and sinks and heat transfer to and from the gas phase of the computational domain. The energy sources are the fi re itself as well as other potential sources that may be present in the simulation such as appliances, radiant/convective heat sources (stoves, radiators). Energy sinks within fi re modelling can be evaporative cooling (sprinklers, water mist) and heat transfer to surfaces. Energy is transferred from the gas phase by convection to surfaces, by advection out of the domain, and by radiant transfer.

4.1 Combustion modelling

As mentioned in Section 1, combustion, even for the simplest fuels, will involve a multitude of chemical reactions and intermediate species and free radicals. For example, hydrogen combus-tion in air (a nitrogen−hydrogen−oxygen mixture) has 82 reactions [13]. While there are times when one desires to model this level of complexity, for a fi re simulation, where the detailed chemistry is not critical, a much lower level of detail is advised. Within FDS, two combustion models are used: a single-step Arrhenius model and a single-parameter mixture fraction model.

In the Arrhenius reaction, separate species are defi ned for fuel, nitrogen, oxygen, and the desired major and minor combustion products. Note that each species will require its own conservation of mass equation. The equations to model the combustion are the chemical equation




for the combustion of the fuel and a reaction rate equation. The equations are shown below where F is fuel, O is oxidizer, P are products, a and b are stoichiometric coeffi cients, C is concentration, B and E are Arrehnius coeffi cients, R is the ideal gas constant, and T is the temperature. Since only one chemical equation is used, the model is referred to as a single-step model.

,aF bO P+ → (7)

a bde .

d

EF RT

F OC

BC Ct

−= −

(8)

The mixture fraction model in FDS is a single-step model which assumes that the chemistry is infi nitely fast and that it always occurs regardless of the local temperature. This phenomenon is succinctly summarized in the phrase ‘mixed is burnt’. That is fuel and oxidizer cannot coexist. Reformulating the single-step reaction allows for either excess fuel or excess oxidizer,

min , max 0, max 0, .

A B B AAF BO P A a F B b O

a b b a + → + − + −

(9)

Since the chemistry is infi nitely fast, only the right-hand side can appear in the computational domain rather than the original unreacted mixture of fuel and oxidizer. In the original unreacted mixture, if one considers any given volume of gas for which the temperature is known, then knowing the quantity of fuel determines the quantity of oxidizer and vice versa (i.e. any mass in a location that was not originally fuel must have been originally oxidizer). Therefore, these two assumptions lead to the conclusion that all possible combinations of products can be determined by a single parameter: the original fuel or oxidizer mass fraction. Thus, by defi ning a term called the mixture fraction, Z, that represents the amount of mass in location that was originally fuel, a series of state relations (such as the one given in Fig. 1 for hydrogen and air) can be generated.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mixture Fraction (Z)

Mas

s F

ract

ion

(kg/

kg)

N2

O2

H2

H2O

Zf

Figure 1: State relationships for hydrogen−air combustion. Zf denotes the location of the fl ame surface.




This concept works surprising well for well-ventilated fl ames as can be seen in Fig. 2, which shows state relationships for methane plotted along with measured data from a Wolfard−Parker slot burner [14−16]. It also works fairly well for data collected during compartment fi res [17] (Fig. 3). However, in the compartment fi re case, the infi nitely fast, mixed is burnt assumption cannot capture the full variability of CO and CO2 in underventilated cases.

In the mixture fraction approach, combustion occurs wherever fuel and oxygen meet. Since the two cannot coexist, the only location at which this occurs is the stoichiometric surface where

0.00

0.05

0.10

0.15

0.20

0.25

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20


Mas

s Fr

actio

n (k

g/kg

)

CH4

O2

N2

H2O

CO2

Figure 2: Methane−air state relationships and slot burner data.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 0.05 0.10 0.15 0.20 0.25 0.30


Mas

s Fr

actio

n (k

g/kg

)

C3H8O2N2CO2H2OCO

Figure 3: Propane−air state relationships and compartment fi re data [17].




the post-reaction mass fractions of both fuel and oxygen are zero. The mixture fraction value at this surface is Zf, as was shown in Fig. 1. At this surface oxygen will exist on the lean side and fuel will exist on the fuel rich side. The HRR is then a function of how quickly the two can meet. The transport of fuel and oxygen is shown below in the transport equations for mixture fraction and oxygen.

,

DZD Z

Dtr r= ∇⋅ ∇

(10)

2 2

2 .O ODO

D Y mDt

r r= ∇⋅ ∇ + ′′′

(11)

By applying the chain rule to the oxygen derivative and summing the equations, a single equa-tion is obtained that related the change in oxygen to the mixture fraction. This equation is shown below.

2 2

2

d d.

d dO O

O

Y Ym D Z D Z

Z Zr r− = ∇⋅ ∇ − ∇⋅ ∇′′′

(12)

This particular formulation, while analytically correct, is not well suited for a numerical solution. However, consider that the entire computational domain can be divided into two domains, a fuel

rich domain where there is no oxygen ( dYO2

___ dZ = 0 ) and a fuel lean domain. The divergence theorem

can be applied to eqn (12) to convert it from a volume representation to a surface representation. This results in:

2

2

d.

d f

OO Z Z

Ym D Z n

Zr == − ∇ ⋅′′

(13)

In the above equation, the mass loss rate of oxygen is determined by the rate at which mixture fraction diffuses across the stoichiometric surface. This is equivalent to saying that the oxygen is consumed as fast as it can enter the fl ame. Oxygen consumption is then related to the HRR by the heat of combustion per unit mass of oxygen. The above equation is advantageous numerically as it only has a single fi rst-order, space derivative. Furthermore, this equation only needs to be evaluated at grid cell faces that separate a fuel rich cell from a fuel lean cell (indicating the fl ame sheet cuts through one of the two cells).

This approach has some limitations to it, though. Two limitations are numerical. A third limi-tation is physical.

The HRR is a function of the gradient. Therefore, if the gradient is poorly resolved, then the local heat release will also be poorly resolved. Although conservation of fuel mass guarantees the right global HRR, the local value of q′′′ is grid dependent. In other words, dq V′′′∫ can be expected to be accurate even though ( , )q x t′′′ is a sensitive variable. This fact can result in two outcomes. Overly large gradients, as might occur near a burner that is spanned by a small number of grid cells, result in local HRRs that exceed those seen in real fi res. Excessive local heat release leads to locally elevated temperature levels which subsequently impact both the buoyancy and the radiation source terms. On coarse grids, the mixture fraction is smeared out by excess artifi -cial diffusion, leading to lower fl ame heights and increased local q′′′ . A related limitation is that a coarse grid will result in the fl ame sheet occupying too small of a volume. This will result in lower fl ame heights which will again lead to buoyancy and radiation source term errors.

The fl ame height error can be corrected by selecting a different surface to integrate over. Since

mixture fraction is a conserved quantity, any value of Z can be selected, provided that dYO2

___ dZ is




appropriately adjusted, and the same heat release will result, provided of course that the new surface lies within the computational domain.

Excessive local HRRs can also be accounted for. There is an upper bound to the amount of energy per unit area of a diffusion fl ame. This upper bound is in part a function of the type of fuel being burned. FDS assumes, with the option for user modifi cation, that combustible materials typical to the built environment are to be used. The assumption is then made that Heskestad’s fl ame height correlation [18] is an adequate characterization of the minimum volume occupied by fi re. If it is further assumed that the shape of the fi re is a cone then the surface area of the fi re can be derived from Heskestad’s correlation. The end result of combining the cone area with the fi re size is a HRR per unit area.

By default FDS will use both of the aforementioned corrections. Using the grid size, the heat release per unit area is converted to a volumetric HRR during the initialization routines. The Z value of the fl ame sheet is updated at each time step based on the resolution of the fi re size (i.e. as the fi re grows larger, the effective grid resolution increases). The volumetric heat release in each grid cell is computed using the effective Z value. Heat release in any grid cell exceeding the volumetric limit is clipped with a running tally made of the clipped heat releases. After all cells are computed, the total clipped heat release is then added to those cells with a non-zero HRR proportional to the heat release of that cell.

4.2 Heat transfer

Heat transfer within a fi re model is an important phenomenon that can have a profound impact on all the solution variables. Surface heating by convective or radiative heating can result in the ignition of other materials. Conduction into an object can result in ignition on another face of the object or it can result in a failure of the object. The removal of heat via water evaporation can act to mitigate the effects of/suppress a fi re.

4.2.1 Convection heat transferConvection heat transfer to a surface is determined by how fast energy can diffuse from the free stream of a fl ow fi eld, through the boundary layer, and into the surface (see Fig. 4).

In a DNS computation, where the boundary layer is resolved, the heat transfer is determined by applying the conduction equation to the temperature gradient normal to the wall:

gas gas wall( ) | .q k T n= ∇ ⋅′′

(14)

In an LES computation, the boundary layer is not resolved. Therefore, the temperature gradient at the wall does not refl ect the actual heat transfer that is occurring. A correlation must be used, therefore, to compute the heat transfer. Correlations for convective heat transfer are typically divided into two categories: forced convection and free convection. While fi re driven fl ows result from free convection, sizeable velocities can be achieved and a forced convection correlation may be more appropriate. To account for this, FDS computes the heat transfer assuming both forced and free convection and then picks the larger of the two. The correlations for the heat transfer coeffi cient [19] are given below with the free convection on the left:

1 4 1

3 5 3max | | , 0.0037 .k

h C T Re PrL

= ∆

(15)




Since the Reynolds number is proportional to the characteristic length, L, the forced convection is only weakly dependant on length. To simplify the computation, L in FDS is assumed to be 1 m.

4.2.2 Conduction heat transferThe general equation for time-dependent heat transfer in a solid material is given by:

1.

Tk T q

c tr∂

∇⋅ ∇ + =′′′∂

(16)

Here, r is the wall density, T is the wall temperature, c is the wall specifi c heat, k is the wall ther-mal conductivity, and q′′′ represents sources terms that can be obtained from phenomena such as pyrolysis. For many of the materials common to building construction, either k, the conductivity, tends to be fairly small (wood, masonry) or the thickness of the material is very thin in com-parison to its exposed surface area (sheet metal). Thus, in general, the primary concern is heat transfer into a material as opposed to across a material’s face. The above equation can therefore be simplifi ed to transfer across the dimension normal to the surface:

1.

T Tk q

x x c tr∂ ∂ ∂

+ =′′′∂ ∂ ∂

(17)

The above equation is used by FDS to model conduction heat transfer. The general approach used is to discretize the surface into a number of nodes and solve the resulting set of linear equations

TW

TB

v→

Figure 4: Turbulent convective heat transfer to a surface is characterized by a parallel average velocity component that increases with distance away from the wall. In this example, the temperature profi le decreases as the perpendicular coordinate moves away from the wall. The opposite trend is possible, the occurrence being problem specifi c.




using a Crank−Nicolson scheme in time. To reduce computational expense, if the material thick-ness and anticipated heat fl ux are small enough, then one can consider the material to be isother-mal in space. This is the lumped mass or thermally thin approximation. This approximation is appropriate to use when the Biot number,

,

h xBi

k

∆=

(18)

is less than 0.1.For a fi re simulation, the heat transfer coeffi cient, h, represents the effective coeffi cient result-

ing from the maximum combined radiative and convective heat transfer that might occur to a sur-face. For example, if a 1500 K gas temperature and 300 K surface temperature are assumed, an effective h of 250 W/(m2·K) is computed. The ratio ∆x

__ k < 0.0004 (m2·K)/W can be used to deter-mine if a material is thermally thin or thermally thick. Hence a 1 cm slab of steel, for which k = 50 W/(m·K), is thermally thin but a 0.1 mm sheet of hardwood, with k = 0.1 W/(m·K), is thermally thick.

Solution of the heat conduction equation for a surface requires specifi cation of the BCs on each side of the surface. These are given by the net convective and radiative fl uxes to the surface as discussed in Sections 4.2.1 and 4.2.3. To keep the solution of the equation stable, especially for thin insulating materials, it is desirable that the heat fl uxes used as BCs at the beginning of a time step still be valid at the end of a time step after the surface temperature has changed. The convective heat fl ux is driven by the temperature difference between the surface and the gas adjacent to it. It is therefore linear with surface temperature as is the conduction equation. It is trivial to incorporate this into the formulation of the BC. The radiative fl ux, however, depends on the difference of the fourth power of the gas and surface temperatures.

The net radiative fl ux is given by the following equation:

4

r,net w r,in w w(1 ) .q q Ts s= − −′′ ′′ (19)

The fourth power dependence can be removed by a Taylor series expansion Tw over a time step. So that Tw can be given as:

1 4 3 1

w w w w( ) 4( ) ( ) ,n n nT T T T+ += −n (20)

where n + 1 and n are time steps.

4.2.3 Radiation heat transferIn Chapter 7 it was established that the radiation transport equation (RTE) is

ss

4

( , )( , ) [ ( , ) ( , )] ( , ) ( , ) ( , ) ( , )d .

4

xs I x s x x I x s B x s s I x sl l l

s lk l s l l

π

⋅∇ = − + + + Φ Ω′ ′π ∫

(21)

Here s is the direction vector for the radiation pencil, Ω is the solid angle, Il is the intensity

in units of power per wavelength per steradian, k is the mass absorption coeffi cient, ss is the scattering coeffi cient, B is the spontaneous emission term, and Φ is the phase function. This equation states that the local gradient of radiant intensity for a specifi c wavelength in a specifi c direction (the left-hand side) is the sum of the loss by absorption and scatter, the gain by local emission, and the gain by in-scatter from other directions. A number of approaches exist to solve this equation.




One approach is to randomly pick at each time step a subset of surface and volume cells, com-pute a number of rays of emission in random directions, and follow the ray until it impacts a surface or leaves the domain. If enough cells are selected and rays generated, summing the results for each ray will result in a reasonable approximation of the solution. This method is referred to as a Monte-Carlo method. If enough random samples are made, it will yield very accurate results including properly capturing the phenomenon of shadowing. For a steady-state computation or computing surface-to-surface view factors, Monte-Carlo methods are computa-tionally affordable. For a complex, multidimensional, time-dependent simulation with a volu-metrically distributed radiative source (fi re, surfaces, and hot gas layer), Monte-Carlo methods become computationally expensive.

A second approach is to utilize, Legendre polynomials to simplify the angular dependence. The simplest such use is the P1 approximation, which uses the fi rst two Legendre polynomials, P0(x) = 1 and P1(x) = x. Applying the P1 approximation to the angular dependence of the RTE, converts eqn (21) to a diffusion equation where radiation is diffused throughout the computational domain. This formulation can be advantageous over the Monte-Carlo method as one simultaneously solves for the entire domain. Therefore, the computational expense of solving the diffusion equation can be less than the expense of the Monte-Carlo method, when the radiant source geometry is complex (e.g. would require a large number of rays to resolve). The main disadvantage of this method is that it diffuses radiation; therefore, shadowing is not handled well as radiation streaming past an obstacle will diffuse around the backside of it.

A third approach which attempts to capture the ray-like behaviour of radiation using a compu-tationally affordable algorithm is the fi nite volume method (FVM). First, the RTE is simplifi ed by assuming a non-scattering gas and by limiting the wavelengths to a small number of bands:

,( , ) ( )[ ( ) ( , )], 1, , ,n n b n ns I x s x I x I x s n Nk⋅∇ = − = … (22)

where n indicates a band and Ib is the source term given by:

4

( )b,n nT

I x Fs

=π

(23)

where Fn is the fraction of the source term emitted into band n. These simplifi cations have elimi-nated the scattering integral and have also limited the wavelength spectrum from a large number of discrete wavelengths to a small number of bands. However, there is still the direction vector which spans the 4π spherical angle. If the direction vector is grouped into a fi nite number of control angles (like wavelengths were grouped into bands), dΩl, and the resulting equation is integrated over each grid cell, then the following is obtained:

( , ) ( )[ ( ) ( , )], 1, , .l lV Vijk ijk

n b,n ns I x s x I x I x s n NkΩ Ω

⋅∇ = − =∫ ∫ ∫ ∫ …n

(24)

Using the divergence theorem, the volume integral on the left-hand side can be replaced by a surface integral over each of the six faces of the grid cell. This resulting equation is:

6

, ,1

ˆ( ) [ ] d , 1, , .m,n

l

l l lm m b ijk b ijk ijk ijk

m

A I s n I I V n Nk= Ω

⋅ = − Ω =∑ ∫ …

(25)

This is the equation implemented in FDS. A number, l, of angles are selected which is typically about 100. To update the radiative intensity, the radiative solution for each angular direction for each band is solved and the results summed over each band to obtain the net radiative intensity.




Since scattering has been ignored, all that matters along any given direction is the net radiation that is transmitted through a grid cell. The solution method, therefore, is as follows. For each angle the upwind corner of the computational domain is determined. The grid is then swept towards the downwind direction. For each grid cell the radiant intensity in that cell is the sum of the intensities from upwind cells, plus any radiative source term in that cell, minus any loss due to absorption. This process is repeated for all angles and all bands.

It is clear that this process involves a computation that loops over the entire domain many times. This can become computationally expensive. Two things can be done to reduce the com-putational expense of this method. The fi rst is to reduce the number of bands. This can typically be done without a signifi cant impact on the radiation solution. For combustion products the spe-cies of interest are CO, CO2, H2O, fuel, and soot. The frequency dependence for these species can be collapsed to 6 or 10 bands depending on the importance of radiant absorption by the fuel. For many common combustible materials, soot dominates the radiative emission and absorption which allows a single band to be used. The second is to reduce the number of angles. However, since radiation is only transmitted along the angles, too few angles and the solution will show signifi cant ‘hot’ spots along the angles and signifi cant ‘cold’ spots between the angles. There is, however, another method to effectively reduce the number of angles, and that is to only update a subset of the angles at each time step.

Consider a typical CFD simulation of a compartment fi re. In general, the geometric distribu-tion of temperature and heat release changes slowly; at least in comparison to the time step size. Under these conditions the radiation source term does not vary greatly and the solution to the RTE also does not vary greatly in time. Thus, only updating a portion of the angles in each time step taking a small number of time steps to fully update the RTE (in effect reducing the number of angles) will not have a signifi cant impact on the solution accuracy. For example, for dt ≈ 0.01 s, updating the radiation calculation every ~0.1 s is adequate.

Solution of the RTE requires determining the absorption coeffi cient, k, and the radiative source term, Ib. For a grey gas the source term is typically given as ksT 4. When a coarse grid is used, one that does not resolve the fl ame, the gas temperature inside of a grid cell with combustion is not likely to be representative of a fl ame temperature. With the fourth power dependence, using a lower tem-perature can result in greatly underpredicting the radiative source in grid cells with combustion. To avoid this FDS uses ksT 4 in all grid cells without combustion. In grid cells with combustion it com-putes both ksT 4 and a radiative source term given by the local HRR multiplied by a user specifi ed radiative fraction, and then FDS uses the larger of the two as the source term. k is function of the local species mass densities. Since species are uniquely determined by the mixture fraction, k is therefore a function of the local Z and T and the frequency band. k is precomputed using RADCAL [20] and stored in a table.

5 Liquid sprays

FDS includes the ability to model the effect of water or fuel sprays. This is done using a Lagran-gian superdrop model. A Lagrangian model follows the path of each droplet individually as opposed to an Eulerian model which would represent all drops using a scalar quantity much like gaseous species are handled. The superdrop indicates that each droplet being track represents a much larger number of droplets. Since a sprinkler nozzle may discharge tens of thousands of droplets per second, to track every droplet would be computationally prohibitive.

A liquid spray of either fuel or water will impact each of the major CFD conservation equa-tions. Evaporation of liquid from the drops and attenuation of radiant heat transfer by the spray




will impact the energy equation. Lastly, the evaporated mass must be accounted for in the mass equation.

5.1 Drop size distribution

Based on the fi ndings of Chan [21], FDS uses a Rosin−Rammler/log-normal distribution to determine the size of the individual droplets coming out of a nozzle. The formula for this cumu-lative size distribution is

– ln2

ln[ / ]11 erf if (Log-normal),

2 2( )

1– e if (Rosin-Rammler).m

mm

d

dm

d dd d

F d

d d

g

s

+ ≤ = >

(26)

Note that the Rosin−Rammler term is identical to

( ) 1 2 .m

d

dRRF d

g − = −

(27)

The operator erf() designates the error function [22]. Figure 5 shows experimental data for F(d). Whenever data like this is available, a non-linear least squares solver can be used to determine g, s, and, if need be, dm. Profi les for F(d), however, are rarely provided by the manufacturers.

d (µm)0

0

100 200 300 400 500 600 700 800 900

100 200 300 400 500 600 700 800 900

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

F(d) RR/LN DistributionF(d) Weighted DataU(d) from RR/LN DistributionC

umul

ativ

e vo

lum

e fr

acti

on F

(d)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Cum

ulat

ive

num

ber

frac

tion

U(d

)

Figure 5: Cumulative volume and number fractions for the Securiplex Velomist water mist nozzle derived from the Rosin−Rammler/log-normal distribution.




Increasingly, though, the drop diameter when F = 0.1, designated Dv[10], the drop diameter when F = 0.5, (Dv[50] or dm), and the drop diameter when F = 0.9, (Dv[90]) are being published. These values, along with eqn (26), suffi ce to determine g, s, and, dm. For the Securiplex Velomist water mist nozzle shown in Fig. 5, the mean droplet diameter, dm, was 175 µm, d = 50 µm when F = 0.1, and d = 400 µm. From these data, along with inverses of eqn (26), it was determined that g = 1.45 and s = 0.978. The results are plotted in Fig. 5. The theoretical Rosin−Rammler/log-normal distribution closely resembles the experimental one that accompanies it.

FDS actually determines the diameter of each introduced droplet from the cumulative number fraction, U(d). It is defi ned as

0

( ) ( )d ,d

U d f d d= ∫

(28)

where f(d) is the probability density function,

33

0

( )( ) .

( )d

F df d

Fd

d dd

∞′=′∫

(29)

The expression for U(d) can now derived. The derivative of the cumulative size distributions is

2ln[ / ]

2

1 ln2

1e if (Log-normal),

2( )

ln 2 e if (Rosin–Rammler).

m

m

d d

m

d

dm

m

d dd

F d

dd d

d

g

s

g

g

s

g

−

− −

≤ π=′ >

(30)

The integral of this function that forms the basis for U(d) is

22

9 / 2

3

3 3 ln23/

3

3 ln[ / ]e erf

2if ,

2( )d

33[ln 2] , ln 2 e

if .

m

m

mm

d

m m

m

d

d ddF

d dd d

g

s

dgg

s ds

dd

d d d gg

g d

−

+ ≤′ =

Γ − −

>

∫

(31)

The operator Γ(,) denotes the incomplete gamma function [22]. Evaluated over the whole range of possibilities, the result is

2ln2 3/

9 / 2

3 3 30

3e 3[ln 2] , ln 2

( ) e 3d erf 1 .

2 2m m

F

d d

gs g

gd sd

d g

−∞

− Γ − ′ = + +

∫

(32)




Making the substitutions, it is evident that the cumulative number fraction is,

2

2

29 / 2

9 / 2 ln2 3/

3 ln[ / ]e erf 1

2( ) for ,

3 2 3e erf 1 e 3[ln 2] , ln 2

2

m

m

d d

U d d d

s

s g

ss

s gg g

−

++ = ≤

+ + − Γ −

(33)

and,

2

3 ln23/

3

ln2 3/9 / 2

3 3

33[ln 2] , ln 2 e

( ) 1 for .3

e 3[ln 2] , ln 2e 3

erf 12 2

m

d

d

m m

m

m m

d d

d d

dU d d d

d d

g

gg

gs

gg

g

ggs

g

−

−

Γ − −

= + >

− Γ − + +

(34)

Equations (26), (33), and (34) are plotted, along with the experimental data for F(d), in Fig. 5. Clearly the Rosin-Rammler/log-normal distribution provides a good fi t to the experimental data. Note how quickly U(d) levels off. In a large enough sample, FDS would provide the correct mix of large and small droplets. The large droplets tend to dominate the trajectories while the smaller droplets have more rapid evaporation. Choosing droplets solely based on cumulative number frac-tion is insuffi cient for water mist in a numerical method that transports far fewer droplets than is actually the case. In reaction to the fi ndings of Hunt et al. [23], NIST implemented a bin selection algorithm in order to introduce more large droplets at each injection cycle.

5.2 Spray pattern creation

The spray pattern from a nozzle is fully defi ned by a spherical drop size distribution, a spherical mass fl ux distribution, and a spherical velocity distribution. That is if one envisions a sphere draw-ing around the nozzle at some distance from its orifi ce (to allow the liquid stream to break into drops), at any given point on that sphere there will be a mass fl ux, a drop size distribution, and a velocity vector. At the dark patches in Fig. 6, droplets are injected at the fl ow rate fraction of the user-specifi ed frequency. The user specifi ed radius for the sphere is employed to convert the fl ux specifi ed at a face to the corresponding fl ow rate through that face. This approach allows data obtained from the detailed diagnostic methods of Putorti et al. [24] to be directly translated into a numerical characterization of the spray. It has also been effective in recreating volumetric fl ux maps obtained in experiments where the water streaming from the nozzle was collected in a uniform array below the nozzle [25]. This fl exibility is most welcome when spray fi res are to be modelled, as some models have non-circular orifi ces.

5.3 Spray momentum

The equation of motion for a drop is obtained by applying the conservation of momentum to the drop. That is the change in velocity for a drop is given by applying the force of gravity and drag




to the drop. Since the gas around the drop may also be in motion, this must also be accounted for. This results in the following equation where CD is the drag coeffi cient:

2d d d D d d d

d 1( ) ( ) .

d 2m u m g C r u u u u

tr= − π − −

(35)

Evaluating CD is key to properly accounting for changes in a drop’s velocity. This is not neces-sarily a simple task. For a single hard ball sphere falling through the air, well accepted correla-tions for CD exist. A water spray, does not, however consist of isolated single hard ball spheres. Water drops can have shapes ranging from near spherical to pancake like shapes depending on the drops size and velocity, that is whether or not surface tension pulling the drop into a sphere is stronger than the drag force attempting to pull the drop apart. Furthermore, a droplet from a spray nozzle is surrounded by many other droplets. Since, the fl ow fi eld resulting from fl ow over a single sphere is different than the fl ow fi eld that results from fl ow over a group of spheres of varying size, the drag forces will be different as well. FDS uses a hard ball drag force correlation which does not presently account for either multiple drop effects or drop shape effects.

Figure 6: A nozzle or sprinkler is characterized by the faces of the user-specifi ed number of solid angles. In this example, the darkened solid angles were proposed to numerically character-ize a high pressure multiport water mist nozzle.




5.4 Droplet heat transfer and evaporation

A liquid drop will exchange mass and energy with its surroundings. It can absorb and emit radiant energy, it can convect energy to or from the gas around it, and it a can add or remove energy through mass exchanged by evaporation or condensation.

Radiative absorption by the droplets is determined in the FVM radiation solver. During initial-ization, the MieV code is used to compute absorption and scattering cross sections as a function of droplet radius and wavelength. In the radiation solver the droplet absorptivity is determined by doing a table lookup using the average drop radius and average drop density for each grid cell. The resulting total absorptivity is added to the gas absorptivity used in the FVM solver. After solving for the new radiative intensity, the droplet absorptivity is then used to determine the amount of radiant energy absorbed by droplets in each grid cell. This quantity is then transferred to the individual drops in a gas cell weighted by surface area. Re-radiation from the drops is not accounted for.

Convective heat transfer to the drops is computed within the droplet update routine. A heat trans-fer correlation, which is a function of the Nusselt number defi ned on the droplet radius, is applied. The actual heat transfer to the drop is then the most limiting of: the heat transfer required to com-pletely evaporate the drop, the heat transfer required to have the drop and the gas in its grid cell be at thermal equilibrium, or the heat transfer given by the heat transfer coeffi cient and time step size.

The droplet evaporation model used in FDS is a quasi-equilibrium model. Droplets will evapo-rate in an effort to reach vapour equilibrium based on the temperature of the gas cell the droplet is located in. Since evaporation will remove energy from the drop that may not be replaced by heat transfer from the gas, in any given time step a droplet may not reach equilibrium with its gas cell. The rate of evaporation is determined using a Sherwood number correlation that accounts for the difference between the droplet equilibrium vapour concentration and the current vapour concentra-tion of the gas cell. Condensation is not accounted for.

Equilibrium vapour concentration is given by the Clausius−Clapeyron relationship:

v

d b

1 1

eq 0e ,

h

R T Tp p

− − =

M

(36)

where the peq is the equilibrium vapour pressure, hv is the heat of vaporization, M is the molar mass, Td is the droplet temperature and Tb is the boiling temperature. The temperature change of a droplet as it evaporates is a function of the energy required to evaporate the drop, the heat being transferred to the drop, and the mass of the drop remaining. If insuffi cient heat is available to maintain the drops temperature, it will cool as it evaporates. This means that the end of time step drop temperature will be lower and it is possible that the evaporation rate given by the Sherwood correlation will result in evaporating too much mass (the actual end of time step vapour pressure being higher than the end of time step equilibrium pressure). Since the equilibrium pressure involves the exponential of the inverse of the drop temperature, a simple analytic solution does not exist. To avoid the potential need to iterate for each droplet in the simulation, a linearized solution to the energy balance is used to limit the droplet evaporation.

Following the computation of heat and mass transfer for a droplet, the gas cell temperature, den-sity, and vapour mass fraction are updated. If the droplet is a fuel droplet, the mass added becomes available for combustion in the combustion routine. Note however, that if the mixture fraction com-bustion model is being used, and the amount of vapour added does not raise the gas concentration above Zf , then no additional combustion will occur.

It is also noted that within the droplet routine, the heat transfer, position update, and evaporation computation are performed sequentially for each droplet. Therefore, in the event that multiple drops




exist in a grid cell, the convective heat transfer and evaporation of the droplets will not be correctly computed. The fi rst drop will see the initial gas temperature and vapour concentration and evapo-rate accordingly, the second drop will see the updated temperature and concentrations, and so on. However, this phenomenon typically occurs in cells near the spray head, where most of the other assumptions break down as well.

5.5 Evaporation impact on divergence

As a drop evaporates it has combined set of effects on the gas surrounding it. The addition of mass to the gas phase is an expansive effect, e.g. it acts to raise the local pressure. Depending upon the drop temperature, local vapour concentration, and local gas temperature, the mass added by the evaporation may be at either a higher or lower temperature which will cause respectively an expansive or contractive effect. This combined effect is added to the divergence equation in FDS as follows:

2

2

H O0 H O

d,

di

ii

YR T Tu m

p tr

g

∇⋅ = + + ′′′

∑ M M

(37)

where the fi rst term in the parenthesis accounts for the net energy exchange between the drop and the gas, and the second term accounts for the mass exchange.

6 Boundary and initial conditions

One of the limits of a CFD simulation is that its extent is bounded in physical space. Even a labora-tory experiment inside a building does not have this stringent a limitation. For example, smoke that escapes the laboratory through a window encounters outdoor fl ows which could lead to the recir-culation of smoke by the same window. Smoke that leaves the computational domain cannot return unless the developer has added a method by which to accomplish this.

The fl ow BC in FDS accommodates fl ows across the outer boundary with either a specifi ed fl ow rate or an open BC based on the pressure difference. Specifi ed fl ow BCs are effective means for modelling exhaust points for forced smoke control systems. Keep in mind that these specifi ed fl ow rate BCs, unlike fans, are insensitive to surrounding fl ow fi eld. The velocity will accelerate or slow down to match the BC.

Each open boundary has a user specifi ed set of conditions for temperature, species concentra-tion, etc. The gas coming in would have these properties. For radiation, the open boundaries are treated as black walls, where the incoming intensity is the black body intensity of the ambient temperature. Open BCs can be used for heat and smoke vents. They do not account for phenom-ena such as vena contracta. In order to capture these effects, the computational domain must be extended beyond the compartment limits, the open BC vent replaced by a wall with an opening, and the now extended boundary will all be open.

FDS has mirror BCs that refl ect the solution. For radiation, the intensities leaving the wall are calculated from the incoming intensities using a predefi ned connection matrix. The modeller needs to use mirror BCs with care since the refl ection process constrains the turbulence of a fi re up against the mirrored BC to a symmetry that it would otherwise not have. The user should also be aware of the implication of a domain with symmetry boundaries. For example, a domain with a fi re in the centre and two orthogonal mirror boundaries actually implies four fi res and four times the calculated HRR. Using a refl ected BC for the ceiling or for the fl oor can be strange indeed. Mirror BCs on opposing walls are to be avoided.




Thermal BCs were covered in Section 4.2. To give the user an idea of the impact of thermal BCs, consider once again the example of a single fi re in a suitably ventilated room. Adiabatic BCs would cause the heat within the compartment to rise too quickly, potentially leading to problems with the radiation transport calculation. Constant temperature BCs (the FDS default) can lead to low upper layer temperatures. Specifi ed material BCs work best. The difference in the temperature fi eld obtained by changing from gypsum to wood wall properties is much less dramatic than would be the case if the change were to adiabatic or isothermal BCs. For this reason, the current FDS algo-rithms for thermal BCs are being improved to provide even more realistic response. The need for dependable thermally decomposing BCs is equally great but, as of this writing, this goal is still in the realm of active research.

When specifying a fi re of known HRR emanating from a surface, make sure that the implied mass loss rate per unit area is consistent with published sources [26]. A large mass fl ux per area can have jet-like dynamics. It will also result in run-time performance degradation because of the CFL condi-tion. An excessively small mass fl ux per unit area will lead to ‘fl ame’ dynamics reminiscent of weakly buoyant fl ows.

FDS requires consistent initial conditions just as differential algebraic equation (DAE) solvers do [27]. The fl ow fi eld and the pressure fi eld are tightly coupled in FDS. Hence every fl ow boundary or internal condition must be consistent with the pressure at that surface. This require-ment is also relevant to the initial conditions. Specifying an initial velocity fi eld without a con-sistent initial pressure will lead to numerical diffi culties.

Often, the best initial state for a series of simulations is obtained by running the simulation for a while to establish prevailing fl ow patterns. Examples include (a) fl ow past a building with openings, (b) stack effects in both buildings and stairwells, (c) contributions from heating, ventilation, and air conditioning (HVAC) equipment, and (d) signifi cant buoyant fl ows coming off devices such as hot plates and running machinery. The goal is to get beyond the initial transients so that the streamlines resulting from the various possible sources fi ll the space(s) in question with a reasonably steady fl ow. From that point on, the items that distinguish one run from another would be introduced into each member of the simulation suite.

7 The practice of modelling

The development of a viable numerical simulation is a recognized diffi cult undertaking. Performing a successful regimen of simulations presents a host of challenges that are often not as well known. It is quite simple to make just one mistake in a simulation that will run for three weeks and obtain results that are useless. Any of the common errors listed in Table 1 can ruin an FDS simulation.

One of the key steps in the realization of a successful modelling project is preparation. Good preparation entails review and research, categorization, focus, and elimination. At the end of this process a set of scenarios should be arrived at which would be the analogue of an experimental test series. The choices must be made in full knowledge of the limitations of the model. Assessment techniques are presented in Section 7.1. Once results are obtained, they should be carefully reviewed for correctness and applicability.

7.1 Preparation

7.1.1 Review and researchAny modelling effort begins with the identifi cation of the problem and an assessment that CFD is an appropriate form of engineering analysis. Typically the modeller will be skilled with one simulation.




The assessment would specify if that model is appropriate for the problem at hand. The next step is the review process. Here, the relevant available information sources are concentrated and examined for pertinent inputs. Typical sources are engineering drawings, fi re and police department reports, eye witness accounts, client requests, applicable government regulations and requirements, military specifi cations, and the various fi re, mechanical, and municipal codes. These usually suffi ce to obtain an initial computational domain. Research is then appropriate in order to supply the remaining miss-ing information and to familiarize the modeller with antecedents. For example, obtaining material properties or fuel combustion data frequently require research of some type. It is not unheard of that experiments were performed in order to obtain the necessary inputs for modelling. The modeller also needs to familiarize herself with the history of the problem. Searches through the technical literature should result in a set of papers where similar experiments and/or simulations were performed and where the relevant basic science is expounded. The rule of thumb is that you, the modeller, should know what to expect before you start the simulation process.

7.1.2 Categorize inputsThe review process will result in a list of concerns. These need to be categorized in order of impor-tance. The technique is analogous with the selection process of parameter based asymptotic expan-sions [28, 29]. In this approach, variables are expanded in terms of a parameter that is large or small in some limit. These expansions are introduced into the governing equations and the limit is taken at the required order. So the process typically begins (ignoring logarithmic terms) with zeroth order terms, then fi rst order terms, etc. In a properly designed expansion, each successive order provides a smaller, corrective addition to the terms preceding it. The important point for modelling is that the variables where introduced together based on their order. So CFD inputs should also be introduced into the computational domain based on their order of impact. For example, fl ow rates from a ventilation system would be included because they are of the same order of magnitude as

Table 1: Common mistakes in FDS modelling.

Problem Manifestation

No fl oor Flow across fl oorInternal open vent Not recognizedVent sign error Smoke not exhaustedPlaced solid surface where vent should be Sealed roomModelled outer walls as obstacles on Numerous outer domain boundaryObstacle (partially) covering the fi re bed Crash or low HRRPlaced sprinkler/detector within obstacle No participation in simulationUsed the default thermal BCs Low temperature fi eldUsed the default emissivity (e = 1) in Incorrect radiative transport enclosure with refl ective wallsSet smoke properties without considering Insuffi cient conservatism that visibility ∝ ∆Hc/YsExtremely coarse computational grid ManyFine resolution in light of the CFL condition Slow run timeDid not make the number of cells in any Slow run time given coordinate a multiple of 2, 3, and 5Did not defi ne enough outputs Wasted simulation




the characteristic velocity of the fi re but the fl ow induced the computer fans would be neglected because they are of lower order. If, subsequently, it is determined that they are necessary, then all fl ows of this order should be introduced into the simulation. Just as asymptotic expansions have a limit beyond which adding extra terms no longer improves the accuracy, the level of input detail for a CFD model is limited by the resolution of the simulation.

7.1.3 Focus and eliminateCFD simulations can become burdened with unnecessary information. Subgrid minutiae are a common example. If subgrid information is important to the simulation, a way must be found to model their impact (see Section 3). This implies a development exercise if the model does not currently accommodate the effect. Frequently, though, a great deal of information will be added about the features of a commercial space that either overlap in a fi xed grid code or lead to a large number of fi ne cells in a code with automatic mesh generation and unstructured grids.

Preparation is the key to focusing the scope and determining what can be eliminated. A hazard assessment of the impact of wind on a burning building provides a good example. The choices for a simplifi ed wind model include direction (three variables), magnitude, thermal stratifi cation, momentum stratifi cation, and wind turbulence parameters (at least four). Clearly a parameter study that varies one variable at a time is not feasible. A better approach would be to identify important zones that, given an unfavourable wind direction, would be affected by the fi re. Then study the wind patterns for the vicinity based on the nearest available soundings. Next, consider the topography of the region. The outcome of this assessment would be a set of weather conditions of interest and another set of secondary importance.

Focus also applies to the goal of the study, including its budget and deadline restrictions. The wind impact study provides another example. The building in question is a paper mill located on the banks of a river in a V-shaped valley that runs from west to east. For years, the mill was isolated, using the river as the chief conduit for raw material (logs) and a service road to transport the fi nished products out of the area. Now, as is the case in many areas, retreat homes are springing up in the valley and the plant’s insurers are requiring the hazard assessment. The consultant’s initial assessment found that westerly and easterly winds provided the greatest hazard. Their impact was quantifi ed via mod-elling. It was also found that the wind perpendicular to the valley tended to trap the smoke in a recirculating pattern that mostly affected the plant. However, a pollution problem was also discov-ered when the plants’ emissions were coupled with the emissions from a suffi ciently large valley population. The last discovery is outside the scope of the initial investigation. The client should be informed of the situation but the matter should not be pursued any further for the present study.

8 Assessing the model, assessing the results

In many ways, a simulation can be like a black box, even to the developer. Several approaches have been developed to assess the limits and capabilities of the model. Verifi cation, validation, and error estimation are the most frequently used techniques. Although verifi cation and validation are synonymous as far as the standard thesaurus is concerned, they have taken on distinct meanings in the fi eld of numerical modelling. Verifi cation is the process by which it is established that the model is implemented correctly and that it is correctly solving the desired equations [30]. Consider the example of a set of quadratic equations to be solved. The goal of the verifi cation exercise would be to determine problems such as the algorithm unintentionally accessing an address outside of the array bounds or ferreting out an improper implementation of the equations, e.g. using b3 under the radical instead of b2. CFD verifi cation exercises are clearly more involved. The procedure




encompasses source code review, collection of a series of know solutions, determination of a series of runs, and quantitative comparison with the exact solutions.

Validation is the process by which it is shown that the simulation is correctly reproducing the desired physico-chemical phenomena [30]. The approach typically involves comparing simula-tion results with experimental data. Ideally, error estimation would produce a priori quantitative assessments of the model’s uncertainties. In practice, this is hard to achieve. What are typically done are sensitivity analyses. These involve the variation of a select group of simulation param-eters in order to assess the impact on the fi nal solution.

The techniques selected so far are of local scope. In other words, the simulation can reproduce the results of an exact solution to a given tolerance, is shown to lie within the error bars of the experimental data, or shown to converge as the grid is refi ned. These outcomes instil a sense of confi dence in the code but no more. As Roache [30] pointed out, these exercises confi rm the appro-priateness of the simulation for the problems that were addressed but they do not prove that the code has been globally verifi ed and validated.

Certifi cation is a process by which a simulation is judged to be appropriate for a given set of problems. The synonyms ‘accreditation’ and ‘quality assurance’ have been used to describe the same process as well. The approaches leading to certifi cation are the same as those introduced above. The certifi cation process is intended to be comprehensive and can take a great deal of effort to achieve.

The US Department of Defense provides comprehensive guidelines the verifi cation, validation, and certifi cation of computer software [31]. This guide provides an excellent introduction into all aspects of the practice. It does, however, cover areas that are not yet relevant to fi re CFD models such as software for automatic control and interactive programs. ASTM E 1355 [32] provides specifi c guidelines for fi re models.

For modelling projects, once a set of results are obtained, they should be subjected to a compre-hensive review process. For example, if the model was used in a fl ow regime where it was not accredited, a careful review of the results is warranted. This could go as far as implementing the techniques of verifi cation and validation. For problems within the scope of the recognized capabili-ties of the model, checking that the solution is within bounds and that the initial modelling assump-tions were accurate may suffi ce.

8.1 Verifi cation

Many journals these days require some verifi cation and/or validation as a condition for publication. As was mentioned before, CFD simulations are often applied to situations not envisioned during the development process. Besides meeting requirements, verifi cation can endow the modeller a sense of confi dence that the CFD code is appropriate for the task at hand. FDS has been shown to reproduce the centreline results for a fi re plume [33]. The approach used is typical of many CFD verifi cations: the results were time-averaged so that a comparison could be made with the analyti-cal equations. The choice of verifi cation exercise should be made as close to the problem at hand as possible. For example, if the problem involves fl ow past a storage tank, a simulation using fl ow past a cylinder is a relevant verifi cation. The wake in Fig. 7 would have to be appropriately time averaged so that it could be compared with the exact solutions published in [34, 35].

8.2 Validation

The goals of validation are the same as those of verifi cation. Typically, though, validation pro-vides the modeller with more options to choose a suitably close example to the problem at hand.




The preparation process should have yielded various relevant references. Often the data published therein makes an excellent choice for a validation exercise. When the budget allows, combining the modelling effort with a series of experiments provides another avenue for obtaining validation data.

As part of a program to certify fi re models for nuclear reactor safety, FDS has undergone a com-prehensive verifi cation and validation process for fi re concerns relevant to that industry. The details can be found in [5, 37]. The FDS theory manual [1] contains a long list of verifi cation and valida-tion publications for FDS.

8.3 Uncertainty and sensitivity analyses

Convergence studies and the more broadly defi ned sensitivity analyses are desirable exercises with often thought-provoking consequences. They are typically not pursued because of the time required. An example of a problem signalling an excessively coarse grid is low maximum tem-perature in a domain with a fi re. Roache [30] gives post-processing techniques that can be used to check the order of a simulation.

Although it is not yet computationally practical to perform a formal uncertainty analysis of a comprehensive fi re simulation such as FDS, the methodology will be outlined. First, consider the algebraic function F which depends on N parameters, Pj,

1( , , , , ), 1, , .j NF F P P P j N= =… … …

(38)

The uncertainty,U , is the standard deviation composed of the terms from the Taylor expansion of eqn (38),

U 22

2 2

1 1 1j j k

N N N

P P Pj j kj j k

F F FS S

P P P= = =

∂ ∂ ∂= + + ∂ ∂ ∂

∑ ∑∑

(39)

Figure 7: Verifi cation exercise showing Karman vortex street past a circular cylinder.




where SPj is the standard deviation of Pj and SPjPk is the covariance of Pj and Pk. The partial derivatives, ∂F/∂Pj, are a vector known as the sensitivity coeffi cients. The relative uncertainty,

U / ,F=u (40)

is used as well. In practice, eqn (39) is truncated to the fi rst term and the measurement or instru-ment uncertainty, ∆Pj, is substituted for SPj. Hence

U 22

2 2

1

.N

jjj

FF P

P=

∂≈ ∆ = ∆ ∂

∑

(41)

For a set of M functions

1( , , ), 1, , ,i i NF F P P i M= =… … (42)

the uncertainty becomes a vector

2

2 2

1

,N

ii j

jj

FF P

P=

∂∆ ≈ ∆ ∂

∑

(43)

and the sensitivity coeffi cients form a tensor.Now consider a system of ordinary differential equations (ODEs),

1 1

d( , , , , , ), 1, , .

di

i M NF

G F F P P i Mt

= =… … …

(44)

Partial differentiation of this system with respect to Pj results in the dynamic sensitivity equa-tions

1

d, 1, , , 1, , .

d

Mi i k i

j k j jk

F G F Gi M j N

t P F P P=

∂ ∂ ∂ ∂= + = = ∂ ∂ ∂ ∂

∑ … …

(45)

In order to determine how the uncertainty varies with time, eqns (44) and (45) are solved simul-taneously. At each time step, eqn (43) is used to determine the uncertainty in Pj.

The spatially discretized equations in FDS can be considered to form a system of DAEs, i.e. a combination of eqns (38) and (42). One possible way of estimating the uncertainty is to proceed as was indicated above, solving the DAE + sensitivity system for each cell and then calculating the uncertainty of at each cell using eqn (43). Although this approach is not comprehensive (for example, the uncertainty in the BCs have not been considered and the uncertainty in the perturba-tion pressure cannot be addressed in this fashion), it does indicate the immense computational scope required for a comprehensive uncertainty analysis of a CFD program. Techniques can be employed to reduce the scope. For example, the uncertainty in the HRR can be quite high. Since the HRR has a profound impact on the results, a one parameter uncertainty analysis can still yield useful results. Typically, one is interested in results (such as temperature) at a discrete set of points. By deriving the sensitivity equations for the corresponding cells, a tractable methodology is obtained by which FDS plus a limited set of sensitivity equations can be used to determine the uncertainty in the temperature at a given number of points.

Sensitivity analyses demonstrate how the solution changes, when one feature of the simulation is varied. Convergence studies have already been presented as an example of grid and/or time




step sensitivity analyses. Another fruitful technique for identifying potential problems is plat-form sensitivity. Figure 8 shows the outcome of running a poorly conceived FDS fl ame spread simulation on four different platforms.

Concepts such as order of accuracy give the modeller an idea of the numerical error that would be expected from a simulation. It is much more diffi cult to estimate how uncertainties in the inputs would manifest themselves in the results. For example, if the HRR is 5 MW ± 0.5 MW, how does this affect the temperature reading at any given place? The formal method briefl y introduced above indicates that considerable development work would be required for a model that does not have native sensitivity analysis capabilities. The question can be answered with no further development by performing a set of simulations with the code at 4.5, 5, and 5.5 MW. Again, as with model devel-opment itself, the parameters chosen for sensitivity analyses must be chosen carefully for budget and scheduling reasons. Ordering the parameters in relevance aids in the selection process. Hamins and McGrattan [38] demonstrate how scaling and empirical relations can be used to obtain good uncer-tainty bounds for certain variables such as the temperature in the upper layer.

8.4 Certifi cation, accreditation, quality assurance

As was mentioned earlier, guidelines for certifi cation can be found in [31]. Some commercial CFD software are accredited by independent agencies such as the United Kingdom Accreditation Service and Lloyd’s Register of Quality Assurance according to quality assurance standards such as ISO 9001. FDS has undergone a variety of certifi cation programmes. The fi rst example occurred when the Alaska Department of Environmental Conservation adopted nomographs resulting from FDS calculations into their in situ oil spill burning decision tree [36]. FDS has been used to determine entrainment rates in balcony spill plumes associated with atrium smoke management systems. FDS results were compared with existing correlations and experimental data and used to develop recommendations for calculating entrainment rates for balcony spill plumes that may be included in future ASHRAE handbooks. FDS is currently being used to investigate the impact of make-up air on fi re plumes in atria with mechanical exhaust systems. The goal of this project is to determine

Figure 8: Heat release rates resulting from running a fl ame spread problem on four different platforms.




the impact of the make-up air velocity on the fi re plume and the hot layer and to determine how large the maximum make-up velocity can be before it can reduce the effectiveness of the exhaust system. FDS passed the SPEC benchmark suite. FDS has been accepted as a hazard analysis tool for the nuclear industry [37].

8.5 Review

The review process for the results of a series of simulations should embody the same rigor as the evaluation of an experimental regimen. All the statistical analysis techniques [39−41] are applicable. The results should be checked for evidence of common mistakes such as those out-lined in Table 1. An important assessment is the adequacy of the assumptions and the approach. Using FDS as an example, the default thermal BC (constant wall temperature) is often used in simulations. If the review process shows lower upper layer temperatures than expected, this con-stant wall temperature should be suspected and new runs performed using a more representative thermal BC are in order.

As has been emphasized repeatedly, the modeller should be particularly wary if the simulation was used outside the limits of its intended development goals or outside of the certifi ed fl ow regime. Checking that the solution is within the bounds of relevant analytical solutions or that the data is suffi ciently similar to that published in the literature is prudent. If none are found then suffi ciently relevant models should be developed that can be tested against published results.

9 Examples

9.1 Grid density

The issue of grid density is often raised in any CFD analysis. One wishes to obtain accurate and converged results which leads one to use more grid cells. However, if one doubles the grid cells in each direction the computational time will increase by a factor of 8, a factor of two for each axis and a fourth factor of two from halving the time step to preserve the CFL condition. This leads one to use fewer grid cells. As an illustration a simulation of a 200 kW pool fi re in the open is performed using a range of grid sizes. From eqn (5), the rule of thumb suggests grid sizes in the range of 5−10 cm. Figure 9 below plots the centreline temperature and velocity profi le for each of the four grids. The plots indicate that at a resolution of 10 cm, that grid is not converged for this fi re. Even at 5 cm, the grid is not completely converged; however, decreasing the grid size 20 % to 4 cm only results in a small change in the centreline quantities. The rule of thumb is a good approximation, but it does not replace the need to perform a proper grid study.

9.2 Turbulence model

In this set of examples, choosing the appropriate turbulence model is examined. Two sets of two cases are simulated with each set being the same except for the choice of turbulence model. Two of the cases model a 2 MW fi re in a 4 m × 4 m domain with 4 cm grid cells. The other two cases are a 20 kW fi re in a 0.4 m × 0.8 m domain with 4 mm grid cells. For all cases the default fuel and surface properties in FDS are used with all boundaries except for the fl oor being open. Contours of HRR per unit volume for each case are shown in Fig. 10. At coarse grid resolutions, running the simulation in DNS mode results in non-physical heat release distribution while the LES mode




results in heat release distribution that like is a snapshot of a diffusion fl ame. At fi ne grid resolu-tions, both modes result in HRR distributions that look like snapshot of a diffusion fl ame. However, the LES result is binary in nature. Combustion appears at essentially one volumetric intensity while the DNS result shows a much wider range of values.

LES DNS

4 mm grid

4 cm grid

Figure 10: Contours of heat release rate per unit volume (kW/m3) for LES and DNS simulations at fi ne (4 mm) and coarse (4 cm) grid resolutions.

Elevation (m)

)s/m(

yticole

Venilret

neC

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

20 cm grid10 cm grid5 cm grid4 cm grid

Elevation (m)

(er

utarepme

Te

nilr etne

C°

)C

0 0.5 1 1.5 2 2.5 30

200

400

600

800

20 cm grid10 cm grid5 cm grid4 cm grid

Figure 9: Plume centreline temperatures and velocities for a 200 kW propane fi re using four grid resolutions.




9.3 Symmetry

There are problems where symmetry will appear to exist. An example of this would be deter-mining detection times or sprinkler activation times for a fi re beneath a fl at ceiling where the distance from the fi re to the walls is large. One would be tempted to treat this problem as a two-dimensional axisymmetric problem or as a three-dimensional axisymmetric problem by using a quarter of the domain with symmetric boundaries. However, symmetry in this case only exists in the time averaged sense. The reality is that the mass fl ow in the plume will rotate about the plume’s centre. This rotation is a signifi cant factor in the entrainment of the plume.

To illustrate this, three FDS simulations were performed. Each simulation is of a plume beneath an unbounded ceiling. The fi rst simulation treats the plume as a 2D axisymmetric prob-lem. The second treats the plume as a 3D problem with symmetry conditions on two sides (one quarter of the domain). The third models the entire domain without any symmetry conditions. The computational domain consists of a 4 m × 4 m × 3 m domain with open sides and 200 kW fi re at the centre of the fl oor. A 7.5 cm grid resolution was used. Along the ceiling at radial dis-tances of 0, 1, and 2 m were a triplet of heat detectors with a 74°C setpoint and response time indexes of 50, 100, and 200 (m/s)1/2. Activation times for the three simulations are shown in the Table 2 below where DNA indicates the detector did not activate.

9.4 Sprinklers

As discussed in Section 5, FDS uses a superdrop, Lagrangian model for sprinklers. There are a number of issues associated with this model that the user must be aware of. These are parameters determining the injection of drops into the fl ow fi eld, the grid size around the sprinkler, and the defi -nition of the sprinkler head’s spray pattern.

The fi rst issue involves the injection of drops into the fl ow fi eld. Three parameters in FDS deter-mine how drops are inserted. These are the time interval between insertions, the number of drops inserted per head in a time interval, and the total number of drops allowed. If the number of drops inserted per time interval is too small, then the mass of the superdrop can become too large and overly perturb the gas fl ow. If the time interval between insertions is too large, the gas cooling and mass fl ow impacts from the water spray will not have a continuous impact. Both will result in a poor resolution of the desired spray pattern.

9.5 Combustible material properties

Surfaces within FDS can be defi ned as combustible. This can be done by a couple of methods. In the fi rst method, the surface is assigned an ignition temperature and a predetermined HRR curve. In

Table 2: Detector activation times (in seconds). DNA indicates the detector did not activate.

0 m distance 1 m distance 2 m distance

50 100 200 50 100 200 50 100 200Case (m/s)1/2 (m/s)1/2 (m/s)1/2 (m/s)1/2 (m/s)1/2 (m/s)1/2 (m/s)1/2 (m/s)1/2 (m/s)1/2

2D 12 7 5 86 43 22 DNA 136 67Quarter 17 9 6 143 71 37 DNA DNA DNAFull 24 13 6 103 55 29 DNA 168 91




the second method, the surface is assigned an ignition temperature and a heat of pyrolysis. In this case the HRR is determined based on the radiative and convective feedback to the burning surface. Using the fi rst method, errors in computing the correct feedback are not a concern; however, apply-ing a single heat release curve globally across the cells of an object may not properly capture the true behaviour of the burning object. Also, the test data used to generate the heat release curve may not be indicative of the thermal environment seen in the simulation. The second method, while having the appearance of greater physical accuracy, is sensitive to the input values and the grid used.

An example of the sensitivity of the FDS burning rate as a function of the inputted material properties and computational grid is discussed below. These simulations make use of the room fi re data fi le that is distributed with FDS. In all simulations the room fi re geometry (a single room with an open door that is fi lled with a variety of furniture shown in Fig. 11) is modifi ed to change the material for all the furniture to the polyurethane fuel defi nition distributed with FDSv4. Four simu-lations are performed. The fi rst three have the same grid defi nition but vary the ignition temperature using 280°C, 285°C, and 290°C. The fourth simulation used an ignition temperature of 280°C, but increased the grid density by 50%.

Figure 12 shows the pyrolysis rates for the four simulations. As can be seen the burning rate behaviour for the 280°C and 285°C ignition temperature cases are very similar; however, the 290°C ignition temperature case shows a markedly different behaviour. The difference of 10°C is only a small percentage of the change in ignition temperature and well within the experimental errors asso-ciated with devices such as the cone calorimeter used to measure the ignition temperature. The denser grid case, while using a 280°C ignition temperature, has burning rate behaviour similar to that of the 290°C case. In general it is currently very diffi cult to predict fl ame spread in a complex geom-etry with any certainty.

Figure 11: Roomfi re4 geometry.




Time(s)

0 100 200 300 400 500 6000.00

0.02

0.04

0.06

0.08

0.10

0.12

280 °C285 °C290 °C280 °C,densegrid

Bur

ning

Rat

e (K

g/m

2 s)

Figure 12: Roomfi re burning rates.

9.6 Radiation solver settings

Since the radiation solver uses a fi nite number of angles, there are preferential directions for radiation heat transfer, which are given by how the angles line up with the computational grid. When the radi-ant source is geometrically large, any preferential heat transfer from a single source cell is washed out when all source cells are accounted for. This is not the case when the radiant source is geometrically small. In Fig. 13 below, are the results of four simulations. In all four simulations the domain is a 4 m × 4 m × 4 m cube with the ceiling open and the remaining surfaces fi xed at 20°C. Two pairs of simulations were run, one with 50 angles in the radiation solver and one with 500 angles. In each pair a 1 m2 and a 4 m2 hot surface was placed at the centre of the ceiling. One would expect to see fi ve hot spots on the surfaces: four slightly below the top centre of each wall and one centred on the fl oor. As shown in the fi gure, the 4 m2 source does show fi ve hot spots for both 50 and 500 angles although some directional bias is shown with only 50 angles. The 1 m2 source with 50 angles shows four hot spots on each wall rather than one while the 500 angle result is as expected.

10 Conclusions

The implementation of an effective computational fi re dynamics simulation entails a balance of computer resources, effi cient deterministic models, fast numerical methods, and effective human interaction. Currently, CFD models such as FDS perform well on smoke transport problems and inconsistently on fl ame spread problems. One of the steps that made the successes possible was the incorporation of submodels to correct for resolution inadequacies. Verifi cation and validation help to identify for developers the areas where further effort needs to be expended. For problems that pass the verifi cation and validation process, the code can move on to certifi cation of the simulation for that application.




For the modeller who is not primarily a developer, the simulation should be used for the prob-lems it was created for and for problems that it was accredited for. Good modelling practice is just as challenging as developing the simulation. Careful preparation and thorough review of results are instrumental in achieving a successful modelling endeavour. The modeller needs to keep in mind that just one error, such as any of those listed in Table 1, can suffi ce to ruin the results.

Acknowledgments

The authors would like to express their gratitude to Prof. M. Faghri of the University of Rhode Island for his excellent editorship and to the staff of the WIT Press for all their support.

References

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Figure 13: Relationship between number of radiation angles and size of radiant source.

50 Angles 500 Angles

4 m2 Source

1 m2 Source




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Fluorescence, Technical Report NIST GCR 03-852, National Institute of Standards and Technology: Gaithersburg, MD, 2003.Trelles, J., Mawhinney, J.R. & DiNenno, P.J., Characterization of a high-pressure multi-[25] jet water mist nozzle for the purposes of computational fl uid dynamics modeling. Com-putational Simulation Models in Fire Engineering and Research, ed. J.A. Capote Abreu, GIDAI: Santander, Spain, pp. 261−270, 20 October 2004.Babrauskas, V., Heat release rates. [26] SFPE Handbook of Fire Protection Engineering, 3rd edn, eds P.J. DiNenno et al., Society of Fire Protection Engineers: Quincy, MA, pp. 3-82−3-161, 2002.Brown, P.N., Hindmarsh, A.C. & Petzold, L.R., Consistent initial condition calculation for [27] differential-algebraic systems. SIAM Journal on Scientifi c Computing, 19(5), pp. 1495−1512, 1998.Kevorkian, J., [28] Partial Differential Equations, Analytical Solution Techniques, 2nd edn, Springer-Verlag: New York, 2000.Van Dyke, M., [29] Perturbation Methods in Fluid Mechanics, Parabolic Press: Stanford, CA, 1975.Roache, P.J., Quantifi cation of uncertainty in computational fl uid dynamics. [30] Annual Re-view of Fluid Mechanics, 29, pp. 123−160, 1997.Department of Defense, [31] Verifi cation, Validation and Accreditation (VV&A) Recommended Practice Guide, Offi ce of the Director of Defense Research and Engineering Defense Mod-eling and Simulation Offi ce, November 1996.ASTM, [32] Standard Guide for Evaluating the Predictive Capability of Deterministic Fire Models, E 1355, American National Standards Institute: New York, 1998.McGrattan, K.B., Baum, H.R. & Rehm, R.G., Large eddy simulation of smoke movement. [33] Fire Safety Journal, 30, pp. 161−178, 1998.Hinze, J.O., [34] Turbulence, 2nd edn, McGraw-Hill: New York, 1975.Libby, P.A., [35] An Introduction to Turbulence, Taylor & Francis: Washington, DC, 1996.McGrattan, K.B., Baum, H.R., Walton, W.D. & Trelles, J., [36] Smoke Plume Trajectory from in Situ Burning of Crude Oil in Alaska – Field Experiments & Modeling of Complex Ter-rain, NISTIR 5958, National Institute of Standards and Technology: Gaithersburg, MD, January 1997.McGrattan, K., [37] Verifi cation and Validation of Selected Fire Models for Nuclear Power Plant Applications, Volume 7: FDS, U.S. Nuclear Regulatory Commission, Offi ce of Nu-clear Regulatory Research (RES), Rockville, MD, and Electric Power Research Institute (EPRI), Palo Alto, CA, NUREG-1824 and EPRI 1011999, May 2007.Hamins, A. & McGrattan, K., [38] Verifi cation and Validation of Selected Fire Models for Nuclear Power Plant Applications, Volume 2: Experimental Uncertainty, U.S. Nuclear Regulatory Commission, Offi ce of Nuclear Regulatory Research (RES), Rockville, MD, and Electric Power Research Institute (EPRI), Palo Alto, CA, NUREG-1824 and EPRI 1011999, May 2007.Wilson, E.B., [39] An Introduction to Scientifi c Research, Dover: Mineola, NY, 1952.Holman, J.P., [40] Experimental Methods for Engineers, 2nd edn, McGraw-Hill: New York, 1971.Taylor, B.N. & Kuyatt, C.E., Guidelines for Evaluating and Expressing the Uncertainty of [41] NIST Measurement Results, Technical Note NISTIR 1297, Gaithersburg, MD, 1994.



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