Chapter 11
Implementation in OZone Table of Contents 1. Introduction.............................................................................. 289 2. Backdraft Combustion Model ................................................... 289 2.1 Model and Sub-Models......................................................... 289 2.2 Basic Assumptions and Principles........................................ 290 2.3 Main Formulation of OZone................................................. 294 2.4. Switch to One-Zone or Two-Zone Models ........................... 296 2.4.1 Zone Model Formulation ................................................ 296 2.4.2 Wall Model Formulation ................................................ 297 2.4.3 Switch Criteria ............................................................... 298 2.5 Mathematical Model ............................................................ 299 2.5.1 Basic Parameters and Input Data .................................. 299 2.5.2 Backdraft Combustion Model......................................... 299 2.5.3 Mass Balance in the Compartment ................................ 302 2.5.3.1 Mass Balance Without Considering the Gas Species. 302 2.5.3.2 Mass Balance Considering the Gas Species............... 303 2.5.3.3 Mass Fraction of Species .......................................... 303 2.6 Gravity Current Speed and Ratio of Mixing ........................ 304
2.7 Backdraft Deflagration Model .............................................. 304 3. Default Values .......................................................................... 305 4. Comments................................................................................. 306
11. Implementation in OZone
1. Introduction The aim of this chapter is to implement in OZone the models developed in this thesis in order to analyse and predict backdraft phenomenon in fire compartments. OZone is a two-zone model developed under the scope of the ECSC research projects “Natural Fire Safety Concept” (NFSC1, 1999) and "Natural Fire Safety Concept - Full Scale Tests, Implementation in the Eurocodes and Development of an User-Friendly Design Tool” (NFSC2, 2000). It is the first two-zone model able to analyse and predict the evolution of backdraft in fire compartments taking into account the geometry of the compartment, the properties of the partitions, the burning material, the dimension of the openings and the gas species accumulated in the compartment and their characteristics (flammable range, explosion properties, etc.)
2. Backdraft Combustion Model 2.1 Model and Sub-Models OZone is composed of one main model and several sub-models. The main model includes: • • •
A two-zone model (compartment and partitions). A one-zone model (compartment and partitions). A model to switch from the two-zone to the one-zone model and from the one-zone model to the two-zone model.
The sub-models are linked to the main model and enable the evaluation of: • • • • • • • •
The heat and mass produced by the fire (combustion model). The mass species production according to the chemistry of burning fuel (chemistry model). The mass transfer from the lower to the upper layer by the fire plume (air entrainment model). The mass accumulation inside the compartment. The heat and mass transfer between the inside of the compartment and the ambient external environment through vertical, horizontal and forced vents (vent model). The flammability limits of the gaseous mixture accumulated inside the compartment according to the gas species present in the compartment (flammability limit model). The temperature and pressure built up inside the compartment in a backdraft deflagration (backdraft deflagration model). The energy delivered inside and outside a compartment in a backdraft deflagration (fireball model).
289
11. Implementation in OZone
2.2 Basic Assumptions and Principles The aim of this section is to introduce the assumptions and principles included in the model. These assumptions are explained below according to a fire’s evolution. A fire starts as a two-zone model. The compartment is divided into an upper and a lower layer. In each layer, the gas properties (e.g. temperature or density) are assumed to be uniform. However, the pressure is assumed to be constant throughout the whole compartment volume (except when evaluating mass exchange through vents). The layers are separated by an adiabatic horizontal plane (at height Zs); see Figure 11-1. They are only connected by an air entrainment model. An air entrainment model is an empirical model that allows the user to estimate the rate of mass drawn from the lower to the upper layer by buoyancy in the fire plume. The plume volume is not taken into consideration (no mass or heat balance is calculated here), so it is thus included in the lower layer volume.
H Upper layer Zs Lower layer Zo Two-zone model (2ZM)
One-zone model (1ZM)
Figure 11-1: Diagram 2ZM and 1ZM. The upper zone is assumed to be opaque and the upper layer partitions (wall and ceiling) are linked to this zone by radiative and convective heat transfer. The lower layer is clear and the lower layer partitions are connected to this zone by convective heat transfer only. Vertical partitions are, thus, divided in 2 parts, one in the lower layer and one in the upper layer. The height of the two parts is equal to the height of each zone, while the height of the upper and lower layers varies. In the lower layer, heat is transferred by radiation from the fire to the lower layer partitions, then by convection from the partitions to the lower layer and by conduction within the partitions. Although the lower layer is clear, the radiation between partitions is not evaluated this is because, on the one hand, temperatures of the different partitions are often quite low and similar, leading 290
11. Implementation in OZone to low radiative heat flux between partitions, and, on the other hand, the radiative heat flux from the fire to the partitions is most often the predominant factor as the flame temperature is relatively high. The fire itself is defined by the pyrolysis rate, the rate of heat release (RHR) and the fire area. Qc is the convective component of the RHR and Qr is the radiative component of the RHR. Qc is often in the range of 0.6 to 0.8 RHR (Karlsson, 2000) and has been fixed in the code to 0.7 RHR. Consequently, the radiative part is fixed to 0.3 RHR. In the model, Qc is transmitted to the upper layer and Qr to the lower layer partitions (through a source term in the lower layer partition formulation). Heat and mass transferred through horizontal, vertical and forced vents are exchanged with the layer at the same height, with some exceptions for incoming air through a vertical vent, which is always added to the lower layer (see Figure 11-1) and for forced vents close to the zone interface. If, during a two-zone model simulation, a switch criterion is met at time ts, the two-zone model is abandoned and replaced by a one-zone model. The switch is made in such a way that the total energy and mass present in the 2ZM system at the time of the switch are fully conserved in the 1ZM system; see Figure 112, (Cadorin, 2003). t ≤ ts
t = ts
t > ts
t > to
t = to
t ≤ to
E2ZM
= =
E1ZM mg ( t)
mU ( t) + mL ( t)
Figure 11-2: Transition from 2ZM to 1ZM In a one-zone model, the compartment is represented as a single zone. In this zone the temperature and density are assumed to be uniform, as is the pressure (except while evaluating mass exchange through vents). The gas in the zone is assumed to be opaque and partitions are linked to the zone by radiative and convective heat transfer.
291
11. Implementation in OZone The fire is defined by the pyrolysis rate, the rate of heat release and the fire area, as in the two-zone model. All mass and energy coming from the fire are added to the single zone. Unlike the 2ZM, no air entrainment model is taken into consideration. Heat and mass transfer through horizontal, vertical and forced vents are exchanged within the single zone. The mass species accumulated in the compartment is calculated by means of the mass balance equations developed in Chapter 5: “Combustion Products in Fires”. The model assumes the species concentration to be uniform in the compartment even if the fire develops in a 2ZM. If no mass species is entered in the model, it calculates, then, the quantity of total unburnt and total burnt mass as a function of the stoichiometric ratio, r. When the flame dies out in the compartment due to a lack of oxygen, the rate of heat release becomes zero but not necessarily the pyrolysis of the fuel. Since the compartment (that is the hot gas) is still hot, thermal mechanisms such as conduction and radiation hit the surface of the fuel, easing the smouldering combustion. The smouldering combustion finishes when the temperature of the fuel is lower than the vaporization temperature or when the entire fire load of the fuel is burnt. All the mass release during this period is considered unburnt. Species concentration is used to check the flammability of the mixture inside the compartment. This level of concentration is checked at every moment from the time at which the fire becomes ventilation controlled until the end of the fire. Two methods are used: •
•
A diffusive method known as Le Chatelier’s rule which indicates if a mixture can become flammable or not by introducing a new supply of air. If the Le Chatelier number is less than 1.0, backdraft is not possible. This criterion is calculated until the time the opening is created. The premixed method developed in Chapter 8: “Flammability Limits of Flammable Mixtures”. This calculates the flammable diagram of the mixture present in the compartment at every moment of time. Then, it determines if the gas species concentration in the compartment is within or above the flammable region defined by the flammable diagram obtained. The risk of is assumed to disappear when gas species concentration in the compartment is found below the lower flammability limit (due to the one-zone assumption, this result must be looked carefully).
The influence of pressure on the flammability limits is neglected. Under normal circumstances, the pressure reached by an under-ventilated fire is quite lower than the pressure that could affect the flammability limits of any mixture. If a gravity current occurs at t0, it is simulated by switching the model from a 1ZM to a 2ZM. As mentioned before, the switch is made in such a way that the total energy and mass present in the 1ZM system at time of switch are fully conserved in the 2ZM system. For that purpose, two initial conditions are 292
11. Implementation in OZone needed: initial lower temperature of 25ºC, and an initial height of the lower layer of 10.0 cm. The speed of the gravity current is evaluated according to the results obtained in Chapter 7: “Gravity Current Prior to Backdraft”. This speed behaves as a function of the geometry and position of the opening, the height of the compartment and the buoyancy parameter. The non-dimensional parameters, v*, is returned indicating the approximate ratio of mixing. The height of the upper-lower layer interface is used to estimate the time at which all the smoke layer will be evacuated. If the mixture is capable of becoming flammable, when the gravity current is created, there will be always a flammable region along the interface of the gravity current ready to be ignited by a hot element. The position of this “hot element” is totally unknown, except in experimental fires. Hot elements are distributed throughout the whole compartment; thus, the ignition of the flammable mixture can happen at any point in space and consequently at any time. The model assumes an ignition and a deflagration from the moment the gravity current starts to flow inside the compartment. From this moment, a parallel simulation starts. On one hand, the fire develops as if no deflagration has occurred (main simulation). On the other hand a deflagration is simulated (secondary simulation) taking as input data those provided by the main simulation (temperature, mass fraction of species, etc.). Figure 11-3 represents this process in a schematic way. Not all the parameters necessary for computing the deflagration are shown.
Main simulation
Temperature evolution Mass fraction evolution t = to Deflagration 1
Secondary simulation
Energy evolution Temperature evolution
Figure 11-3: Parallel simulation: normal fire evolution and backdraft deflagration. The model includes three different positions of the ignition source when calculating a deflagration: close to the vent, far from the vent and in the middle of the compartment. They are automatically chosen according to the position of the gravity current, which has been calculated using the non-dimensional parameter (Fr). 293
11. Implementation in OZone A well-stirred situation after ignition is chosen in the compartment. A uniform pressure and temperature are assumed in the compartment. Duration of the deflagration is short enough in comparison to characteristic heat transfer time from hot gases to walls and obstacles, so the adiabatic approach to the phenomenon is employed (adiabatic compression). Different degrees of combustion are assumed to take place inside and outside the compartment. Thus, stoichiometric combustion is supposed outside the compartment (fireball) and combustion governed by a limiting reactant inside the compartment. A discussion of these default values is given in Section 4: “Default Values”.
2.3 Main Formulation of OZone The compartment can be divided into two zones (upper and lower) or can be regarded as in one zone. In a two-zone situation, the gases in each layer have attributes of mass, internal energy, density, temperature, and volume denoted respectively by mi, Ei, ρi, Ti and vi, where i represents the upper or lower layer. A numerical fire zone model is normally based on eleven physical variables which are linked seven constraints and four differential equations describing the mass and energy balanced in each zone. These seven constraints are indicated below. As for the number seven, this represents the calculation of density, internal energy and the gas law twice (once for each layer):
ρi =
mi
Vi
(11-1)
E i = c V (Ti )m i Ti
(11-2)
p = ρiRTi
(11-3)
V = VU + VL
(11-4)
The specific heat of the gas at constant volume, cv(Ti), and at constant pressure, cp(Ti), the universal gas constant, R, and the ratio of specific heat, γ(Ti ) , are related by Eq. (11-5) and Eq. (11-6): R = cp (Ti ) − c v (Ti )
γ(Ti ) =
cp (Ti ) c v (Ti )
(11-5) (11-6)
The variation of the specific heat of the gas as a function of temperature is taken into account by the following relation obtained by a linear regression using the tabulated data for air (combustion products are thus ignored) given in the SFPE Handbook of Fire Protection Engineering (1995):
294
11. Implementation in OZone cP (T) = 0.187 T + 952
(11-7)
The mass balance equations have the general form of Eqs. (11-8) and (11-9) for the upper and lower layer, respectively. A dotted variable xɺ indicates the derivative of x with respect to time.
ɺU =m ɺ fi + m ɺ U,VV,out + m ɺ U,HV,in + m ɺ U,HV,out + m ɺ U,FV,in + m ɺ U,FV,out + m ɺe m
(11-8)
ɺL =m ɺ U,VV,in + m ɺ L,VV,in + m ɺ L,VV,out + m ɺ L,FV,in + m ɺ L,FV,out − m ɺe m
(11-9)
The energy balance equations have the general form of Eqs. (11-10) and (11-11).
qɺ U = Qc + qɺ U,p + qɺ U,VV,out + qɺ U,VV,r + qɺ U,HV,in + qɺ U,HV,out + qɺ U,FV,in + qɺ U,FV,out + qɺ e (11-10) qɺ L = qɺ L,p + qɺ U,VV,in + qɺ L,VV,in + qɺ L,VV,out + qɺ L,VV,r + qɺ L,FV,in + qɺ L,FV,out − qɺ e
(11-11)
Four basic variables have to be chosen to solve the system. Provided that the zone temperatures, TU and TL, the height of the separation of zones, ZS and the difference in pressure from the initial time, ∆p are selected, Eq. (11-8) to Eq. (11-11) can be transformed (Forney, 1994) into the system of ordinary differential equations (ODE) formed by Eqs. (11-12) to (11- 15):
i
∆p =
( γ − 1) ( qɺ U + qɺ L ) V
( ( qɺ V
(11-12)
ɺ = T U
i 1 ɺ U TU + VU ∆p qɺ U − cp (TU )m cp (TU ) ρU VU
ɺ = T L
1 cp (TL ) ρL
Zɺ S =
i
L
)
ɺ L TL + VL ∆p − cp (TL )m
L
i 1 (γ (TL ) − 1) (qɺU + qɺL ) − VL ∆p γ (TL ) PA f
(11-13)
)
(11-14) (11-15)
In the case of a one-zone model, the number of variables which describe the gas in the compartment as a whole is reduced to six, i.e. the mass of the gas, mg; the temperature of the gas, Tg; the volume of the compartment (constant), V; the internal energy, Eg; the pressure inside the compartment, p; and the gas density, ρg. The number of constraints is then reduced to 4: ρg =
mg
V E g = c V (Tg )m g Tg
(11-16)
p = ρg RTg
(11-18)
V = constant
(11-19)
(11-17)
295
11. Implementation in OZone Eq. (11-20) expresses the mass balance.
ɺg =m ɺ fi + m ɺ g,VV,in + m ɺ g,VV,out + m ɺ g,HV,in + m ɺ g,HV,out + m ɺ g,FV,in + m ɺ g,FV,out m
(11-20)
Eq. (11-21) expresses the energy balance.
qɺ g = RHR + qɺ g,p + qɺ g,VV,in + qɺ g,VV,out + qɺ g,VV,r + qɺ g,HV,in + qɺ g,HV,out + qɺ g,FV,in + qɺ g,FV,out (11-21) Two basic variables have to be chosen to solve this system. Provided that the zone temperature, T, and the difference in pressure from the initial time, ∆p, are chosen, Eqs. (11-20) and (11-21) can be transformed into the system of ordinary differential equations (ODE) formed by Eqs. (11-22) and (11-23). i
∆p =
ɺ = T g
(γ − 1)qɺ g
(11-22)
V 1 cp (Tg ) ρg
i
V
ɺ g Tg + V ∆p) (qɺ g − cp (Tg ) m
(11-23)
2.4. Switch to One-Zone or Two-Zone Models If certain criteria are encountered during a two-zone simulation, the code will automatically switch to a one-zone simulation, which better describes the situation inside the compartment at that moment. The simulation will continue until the end of the fire under the hypothesis of a one-zone model, except when a sudden opening creates a gravity current, at which point the model changes to a two-zone model again. The section below shows how the code deals with the basic variables of the zone models, how it sets the initial conditions of the one-zone model and how it deals with partition models. The criteria for switching are also listed in Section 2.4.3: “Switch Criteria”. A complete description of the criteria and of their effect on the simulation process is given by Cadorin, 2002b.
2.4.1 Zone Model Formulation The values of the eleven basic variables describing the gas in the two zones are known at ts, the time at which the switch from the 2ZM to the 1ZM occurs. To continue the simulation with a one-zone model, Eqs. (11-22) and (11-23) must be solved. In a one-zone model there are six variables describing the gas in the compartment as a whole, linked by two constraints. Thus, two new constraints are needed to fix the new initial conditions.
296
11. Implementation in OZone These two additional conditions are obtained by imposing that during the transition from two zones to one zone, the total mass and the total energy of gas in the compartment are conserved. m g (ts ) = m U (ts ) + m L (ts )
(11-24)
E g (ts ) = E U (ts ) + E L (ts )
(11-25)
The initial one-zone temperature Tg(ts) and one-zone pressure p(ts) can be deduced from Eqs. (11-24) , (11-25) and (11-16) to (11-19). A consequence of this procedure is that Tg(ts) is lower then TU(ts) and higher than TL(ts) and thus the temperature curves are discontinuous at the time of the switch. Now, considering that to is the time at which the gravity current is created, meaning the time at which the switch from the 1ZM to the 2ZM occurs, the same principle of mass conservation and energy is applied. m g (to ) = m U (to ) + m L (to )
(11-26)
E g (to ) = E U (to ) + E L (to )
(11-27)
Eqs. (11-26) and (11-27) must be solved in order to continue the simulation in this situation. Two initial conditions must be introduced that will be re-adapted in order to fulfil the mass and energy conservation condition. One of these initial conditions is the depth of the upper layer, 10.0 cm, and the other one is the temperature of the lower layer, 293K. A consequence of this procedure is that TL(to) rises very quickly at the beginning of the switch.
2.4.2 Wall Model Formulation The partition temperatures, the height of the lower and upper walls (vertical partitions) ZS(ts) and H-ZS(ts) are known at time ts. From ts to the end of the calculation, the one-zone model is linked to the lower and upper walls which keep the dimensions they had at time ts, i.e. ZS(ts) and H-ZS(ts). During the transition no modification of partition temperatures or wall dimensions is made; only the boundary conditions at the interface with the interior of the compartment are modified. This procedure enables the user to respect the conservation of energy fully during the transition from the two-zone to the onezone model and from the one-zone to the two-zone model. With a two-zone model, upper walls (and the ceiling) exchange energy by radiation and convection with the upper layer, considered opaque, while lower walls (and the floor) are heated directly by radiation from the fire, and they give back energy to the lower layer by means of convection. With a two-zone
297
11. Implementation in OZone model, all partitions (upper and lower walls, ceiling and floor) exchange energy by radiation and convection with the single opaque.
2.4.3 Switch Criteria The switch from 2ZM to 1ZM can be activated by four criteria. Two criteria are linked to the ignition of the fuel, while two others are related to the loss of validity of the two-zone assumption. While the first two criteria are related to physical phenomena that may occur during compartment fires, the last two are related to the limits of applicability of the two-zone model assumptions. The switch from 1ZM to 2ZM is only activated when an opening creates an entering gravity current. The criteria for the transition from two to one-zone and/or for the modification of the fire source are: •
Criterion I [ TU > Tfl ]: when the gas temperature of the upper layer, TU, is higher than the temperature that leads to flashover, Tfl.
•
Criterion II [ Zs < H q and TU > Tign ]: when the gases in contact with the fuel have a higher temperature than the ignition temperature of fuel (Tign). Hq is the maximum height of the combustible material and ZS is the interface height.
•
Criterion III [ ZS < a 1·H ]: when the interface height drops to a certain percentage, a1, of the height of the compartment, H.
•
Criterion IV [ A fi > a 2 A f ]: when the fire area is greater than a certain percentage, a2, of the floor surface of the compartment.
Criteria I and II lead necessarily to a modification of the rate of heat release. If the fire load is localised, the simulation will continue using a 2ZM, whereas if the fire load is uniformly distributed, a 1ZM will be used. If one of either criterion III or criterion IV is fulfilled, the code will switch to a one-zone model but the RHR will not be modified, except if criterion I or II happens simultaneously. More information on these criteria can be found in Cadorin, 2003. It is important to note that users have the possibility of activating or not any of these criteria and of deciding on the value of the relevant parameters at which criteria are met. The default values proposed here are only informative and not necessarily appropriate in every situation.
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11. Implementation in OZone
2.5 Mathematical Model 2.5.1 Basic Parameters and Input Data To be able to and/or analyse a backdraft fire, the basic parameters necessary to know are: • • • •
The rate of heat release, RHR(t), in W. ɺ fi (t), in kg/s. The pyrolysis rate m The fire area Afi(t). Mass species released by the burning fuel, g/g, and their characteristics such as flammability range, burning speed, combustion heat, etc.
The rate of heat release is taken into account in energy balance and pyrolysis rate is taken into account in mass balance. The mass species are taken into account in mass balance and are essential for evaluating flammability of the gaseous mixture and the evolution of the deflagration. fire area is used in some air entrainment models in OZone.
the also the The
There are two ways of introducing the rate of heat release, the pyrolysis rate and the fire area. The first one is defined as “Design Fire” and the second one is defined as a “Use-Designed Fire”. The procedure for introducing those curves according to “Design Fire” is done according to the Annex E of Eurocode 1 (EN1991-1-2, 2002). These curves can be modified by the models depending on the ventilation conditions, the feedback heat to the fuel, smouldering combustion and the extinction of flames. In User-Design Fires, these curves are introduced directly by the user and they are considered to represent the real evolution of the fire (no modification in the curves are carried out by the model). Therefore, it is assumed that the data introduced includes the extinction of flames, smouldering combustion, etc.
2.5.2 Backdraft Combustion Model The timeline of the backdraft phenomenon is used here to explain the mathematical model. Figure 11-4 shows a schematic outline of a backdraft development as a function of temperature. There are four key time-variables that are used here to differentiate these stages: • • • •
Time tig is the time at which the fire starts in the compartment. Time tvc is the time from free combustion to combustion with lack of oxygen. Time tsc is the time that marks the beginning of smouldering combustion. Time to is the time of the new supply of oxygen, i.e. the creation of a gravity current.
299
Temperature
11. Implementation in OZone
ti
tsc
tvc c
to
Time
Figure 11-4: Average temperature evolution in a backdraft phenomenon. Stage from time ti to time tvc During this stage, oxygen is still present in the compartment. The fire develops in a fuel-controlled situation and all the mass lost by the fuel releases energy inside the compartment. Also during this stage, the mass lost and the rate of heat release are governed by the following equations:
ɺ fi ( t ) = m ɺ fi,data ( t ) m
(11-28)
ɺ fi ( t ) H f.eff RHR ( t ) = RHR data ( t ) = m
(11-29)
Stage from time tvc to time tsc When all the oxygen initially present in the compartment has been consumed up to a certain concentration level defined by the user, the fire becomes ventilation-controlled (incomplete combustion). During this stage, the rate of heat release is governed by the rate of oxygen coming into the compartment though the vents:
RHR ( t ) =
ɺ ox,in m r
H f.eff
(11-30)
ɺ fi ( t ) , can be considered in two ways: The total mass lost, m
Option I- According to NFSC Design Fire and choosing an external flaming combustion: ɺ fi ( t ) = m ɺ fi,data ( t ) m
(11-31)
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11. Implementation in OZone
Option II- According to the pyrolysis model developed in Chapter 4 for non-charring materials:
ɺ =m ɺ + m " fi
" ∞
σεu ζ ( Tg4 (t) − To4 )
(11-32)
Lv
Stage from time tsc to time to When the oxygen entering through the opening is not enough to sustain the flame, the flame dies out. At this moment, the rate of heat release becomes zero. RHR ( t ) = 0.0
(11-33)
The mass lost is governed by the smouldering combustion of the fuel, expressed as:
Option I- According to NFSC design fire and choosing an external flaming combustion: ɺ fi ( t ) = m ɺ fi,data ( t ) m
(11-34)
Option II- According to the pyrolysis model developed in Chapter 4 for non-charring materials:
ɺ fi ( t ) = m
σε ( Tg4 (t) − Tv4 ) + h ( Tg (t) − Tv ) − λ
( Ts
− Tc (t)) y(t)
Lv
(11-35)
Smouldering combustion will finish when: • •
The temperature of the gas, Tg, is lower than the vaporization temperature of the fuel, or when the entire amount of fuel has been consumed.
Option III- Otherwise, smouldering combustion is considered: ɺ fi,data ( t ) = 0.0 m
(11-36)
Stage from time to From time to, a backdraft deflagration is simulated at every time step taking into account the quantity and kind of gas species accumulated in the compartment. The initial data for each deflagration includes the temperature, pressure and mass species concentration at the time of deflagration.
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11. Implementation in OZone
2.5.3 Mass Balance in the Compartment If mass species are not considered in calculation, the model gives: the unburnt gas, the burnt gas and air. However, if they are considered, the model gives the mass concentration for each species.
2.5.3.1 Mass Balance Without Considering the Gas Species Unburnt balance:
ɺ unburnt = m ɺ unburnt,in + m ɺ unburnt,out m
(11-37)
The initial mass of unburnt gas in the compartment is considered to be 0%. The mass of unburnt gas starts entering the compartment when the fire becomes ventilation-controlled. It is considered to be the total pyrolysis mass minus the portion combusted by the oxygen (see Figure 12-5). The mass of unburnt gas leaving the compartment is considered to be ξunburnt % (see Section 2.5.3.2) of the total mass of gas going out of the compartment. Burnt balance:
ɺ burnt = m ɺ burnt,in + m ɺ burnt,out m The initial mass of burnt gas in the compartment is considered to be 0%. The mass of burnt gas entering the compartment is considered to be the pyrolysis mass combusted by the oxygen multiplied by (1+r); see Figure 12-5. The mass of unburnt gas leaving the compartment is considered to be ξburnt % (see Section 2.5.3.2) of the total mass of gas leaving the compartment. 0.25
[kg/s]
0.2
0.15
ɺ fi (data) m
ɺ unburnt m
ɺ fi m
0.1
0.05
0 0
100
200
300
400
500
600
Time [s] Figure 12-5: Division of the pyrolysis rate: Unburnt and burnt mass.
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11. Implementation in OZone Oxygen balance: The mass of oxygen in the compartment is calculated at each time step by integrating the following balance:
ɺ O2 = m ɺ O2 ,in + m ɺ O2 ,out − r·m ɺ fi m
(11-38)
The initial mass of oxygen in the compartment is considered to be 23% of the initial mass of gas, assumed to be fresh air. The mass of oxygen entering the compartment is considered to be 23% of the mass coming in the compartment through the vents. The mass of oxygen leaving the compartment is considered to be ξox % of the mass of gas leaving the compartment. Nitrogen balance: The mass of nitrogen in the compartment is calculated at each time by integrating the following balance: ɺ N2 = 0.77m ɺ air,in − m ɺ N2 ,out m The initial mass of nitrogen in the compartment is considered to be 77% of the initial mass of gas, assumed to be fresh air. The mass of nitrogen entering the compartment is considered to be 77% of the mass of gas entering the compartment through the vents. The mass of nitrogen leaving the compartment is considered to be ξN2 % of the mass of gas leaving the compartment.
2.5.3.2 Mass Balance Considering the Gas Species Species i balance: The mass of species i in the compartment is calculated at each time step by integrating the following balance:
ɺ i = y im ɺ fi − m ɺ i,out m The initial mass of species “i” in the compartment is considered to be 0% of the initial mass of gas, assumed to be fresh air. Also the mass of species i entering the compartment is considered to be 0% of the mass of gas entering the compartment through vents and yi of the pyrolysis mass. The mass of species i leaving the compartment is considered to be ξi % of the mass of gas leaving the compartment.
2.5.3.3 Mass Fraction of Species ξi is the concentration of species i in the gas inside the compartment. It is calculated using Eq. (11-39):
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11. Implementation in OZone ξi =
mi mg
(1ZM)
(11-39)
The concentration of nitrogen is assumed to be uniform in the compartment.
2.6 Gravity Current Speed and Ratio of Mixing The position and the average velocity of the gravity current in the compartment is obtained according to the non-dimensional parameter, v*, obtained as a function of the opening geometry, compartment ratio, presence (or not) of obstacles in the compartment and the buoyancy parameter between the inner gas and the outer air.
2.7 Backdraft Deflagration Model As was mentioned in Section 2.2: “Basic Assumptions and Principles”, a wellstirred situation is assumed inside the compartment when backdraft deflagration occurs. At this time the model can switch to a 2ZM (if the criterion is activated). In a 2ZM, the input data needed to calculate the deflagration are: • •
The mass fraction of the gas species accumulated at this time. The mass balance is always calculated as 1ZM. The temperature of the gas, which will be chosen as the average temperature between the upper and lower layers.
In a 1ZM, the input data are the average temperature in the compartment and the mass fraction of the gas species accumulated at this time. The main equations that describe a deflagration have been already explained in Chapter 9: “Ignition of the Flammable Region: Backdraft Deflagration”. In this section, the main equations of the model will be written again with the sole purpose of having them all summarized in one chapter. For further details, readers are invited to refer to Chapter 9. The following equations represent the dimensionless pressure, π; the relative unburnt mass that becomes consumed, nu→b; the relative unburnt and burnt mass expelled through the vents, nu→v. χ ( τ ) Zπ dπ = 3π dτ
ε+1/ γ u
(1 − n π
π1/ γ u −
)
−1/ γ u 2 / 3
u
γu − γb nu γu
2/3 dn u→b = 3χπε+1/ γ u (1 − n u π−1/ γu ) dτ
− γ u WR Σ
(11-40)
(11-41)
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11. Implementation in OZone dn u→v = −3R u W ∑ (1 − A j ) µ jFj / ∑ µ jFj dτ
(11-42)
At any moment in time, the dimensional deflagration parameters can be obtained by the following relations: • • •
The time of the explosion in seconds, t = τa/Sui. The current pressure, in Pa, in the enclosure, p = πpi. The current mass of unburnt and burnt gas, nu=mu/mi and nb=mb/mi.
The relative unburnt mass of gas that becomes consumed, nu→b as well as the relative unburnt mass of gas expelled through the vents, nu→v can be expressed ɺ mixt ɺ mixt in moles by merely using the molecular weight of the gas, mol u → b and mol u →v , respectively. These terms are used in the following energy equations in such a way that the energy released inside the compartment, Eback_in, the energy released outside the compartment, Eback_out, and the energy not consumed can be calculated.
E Back _ in =
∑∫
tignition
i
E Back _ out =
tflame
∑∫
ɺ mixt ∆H ic (T)·mol u → b ψ i ξ i dt
tflame
tignition
i
ɺ umixt ∆H ci (T)·mol → v ψ i ξ i dt
E Back _ notconsumed = E Back _max − ( E Back _in + E Back _out ) E Back _ max =
∑ ∆H (t i c
ignition
, T)·mol mixt stored ψ i
(11-42)
(11-43) (11-44) (11-45)
i
These equations form only a part of the equations developed in Chapter 9 and Chapter 10. This deflagration model can be used as a model independent from OZone. By simply introducing the required data at a time t, a deflagration can be calculated.
3. Default Values In this section, a summary of the default values set in the code is given. Most of these values can be modified by the user; nevertheless, others cannot be modified and if so are designated “(fixed)”.
305
11. Implementation in OZone • • • • • • • • •
Convective component of the RHR: RHRc = 0.7 RHR (fixed) Radiative component of the RHR: RHRr = 0.3 RHR (fixed) Oxygen/Fuel stoichiometric ratio: 1.3 (fixed) Discharge coefficient for vertical openings: Cf = 0.7 Emissivity of gas: εg = 1 Emissivity of partition: εw = 0.8 Relative emissivity of partition-gas interface: εp = 0.8 Convective heat transfer coefficient of partition-gas interface: h = 25 on the inner face of partitions and h = 9 on the outer face of partitions “Heskestad” plume model
4. Comments Besides being able to analyse and predict the risk of backdraft in fire compartments, OZone has been provided with models that allow the user to analyse:
Vented deflagration in compartments with single or multiple vents. The vent can be initially opened or closed. The toxic hazard in a compartment according to the generation of combustion products.
Nomenclature not yet defined in other chapters
qɺi ,α , β
[W]
mɺ i,α,β
[kg/s]
energy exchange through vent. i = U or L or g. β = in or out.
α = VV or HV or FV. rate of mass of gas exchange through vent. i = U or L or g. β = in or out.
α = VV or HV or FV.
Subscripts c data FV g HV i
variable variable variable variable variable equal
in L out r U VV
variable variable variable variable variable variable
related to convective heat transfer set in the data of OZone or, in general, used as input of a method related to the forced vent related to the single zone of 1ZM related to the horizontal vent to U for variable related to upper layer, to L for variable related to lower layer and to g for variable related to the single zone of 1ZM related to a quantity added to a compartment zone related to the lower layer related to a quantity subtracted from a compartment zone related to radiative heat transfer related to the upper layer related to the vertical vent
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