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A simplified model for postflashover burning of thermoplastic materials
Ser
TH1
N21d
National Research
Conseil national
no,
1677
1*1
Council Ca,
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BLDG,
Institute for
lnstitut de
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Research in
recherche en
Construction
construction
A Simplified Model for Posfflashover
Burning of
Thermoplastic Materials
by D. Yung
Reprinted from
Heat Transfer and Combustion Systems
presented at the
Winter Annual Meeting of the
American Society of Mechanical Engineers
San Francisco, CA
Dec. 10-15,1989
(HTD-122), p. 45-51
(IRC Paper No. 1677)
NRCC 32347
NRC-
ElSTlL t B R A R Y
B I B L I O T H ~ Q U E
I R C
CNRC-
ICtSTLes kquations r6gissant la dynamique d u feu apres l'embrasement gknkral
d'un incendie faisant intervenir des materiaux thermoplastiques sont
analys6es par similitude. Par souci de simplicit6 et de prudence dans la
mod6lisation d u comportement d'un incendie, les dkperditions de chaleur
2i
travers les cloisons ne sont pas prises en compte. I1 est dkmontr6 que le
comportement du feu est rbgi par deux kquations algkbriques. Les variables
dkpendantes choisies sont la temperature sans dimension des gaz et la vitesse
sans dimension de combustion. I1 s'avere que ces dernieres ne dkpendent que
de six parametres de similitude et de trois coefficients. Deux des coefficients,
le rendement de la combustion et le rendement de 116vaporation, ne sont pas
determinks dans la pr6sente analyse, mais sont fixks B leurs valeurs
maximales pour des raisons de prudence dans le calcul de la r6sistance au feu
et de la propagation des flammes. Des solutions numkriques sont port& sur
graphique. Si l'on suppose que les valeurs des deux rendements sont
raisonnables, le modele produit des pr6visions de la tem@rature des gaz et de
la vitesse de combustion qui se comparent avantageusement aux donn6es
experimentales disponibles.
Reprinted From
HTD-Vol. 122, Heat Transfer in Combustion Systems
@
m e American Society ofEditors: N. Ashgriz, J. G. Quintiere, H. G. Semerjian, and S. E. Slezak
Mechanical Engineers Book No. H00525
-
1989A SIMPLIFIED MODEL FOR POSTFLASHOVER BURNING OF
THERMOPLASTIC MATERIALS
D. Yung
Fire Research Section Institute for Research in Construction National Research Council of Canada
Ottawa, Ontario, Canada
ABSTRACT heats ratio = AHdAH,
The equations governing the fire dynamics of a q efficiency
postflashover fire involving thermoplastic materials are 8 dimensionless temperature (defined in equation (1 0)) analyzed through similarity considerations. For simplicity p air density, kg/ms
and for conservative modeling of the fire behaviour, the heat ~ t ~ f constant ~ ~ = 5.67 10-8 W/m2.K4 - ~ ~ l t ~ ~ ~ ~ ~ loss through the compartment walls is neglected. The fire
behaviour is shown to be governed by two algebraic ventilation factor = pa ~ , . I K k g / s equations. The dependent variables chosen are the
dimensionless gas temperature and the dimensionless burning rate. They are shown to depend on only six
SubscnDts
similarity parameters and three coefficients. Two of the a ambient coefficients, the combustion efficiency and the evaporation c combustion efficiency, are not determined in the present analysis but are e evaporationassumed to be at their maximum values for conservative fire g flames and hot gases resistance and fire spread design considerations. r radiation
Numerical solutions are mapped out. Assuming reasonable v ventilation opening values for the two efficiencies, the model is shown to predict
gas temperature and burning rate that compare favourably INTRODUCTION with available experimental data.
In the course of fire development in a compartment, NOMENCLATURE the fire initially grows primarily through spreading over
combustible surfaces. As the fire develops, it produces a fire
a
radiation area parameter (defined in equation (lo)), plume, which in turn creates a hot ceiling layer of flames and dimensionless hot gases. Thermal radiation from the plume and the hot A area, m2 ceiling layer raises the temperature of combustibles in other C, flow coefficient = 0.1 38 parts of the compartment, eventually reaching the flashover C specific heat, J1kg.K point when all remaining combustibles in the compartment F radiation configuration factor, dimensionless ignite simultaneously. After flashover, the fire develops into g gravitational constant = 9.8 mIs2 a fully-developed, compartment fire called postflashover fire.h height, m Whereas the initial fire development is important for fire detection and suppression considerations, the postflashover AH heat of combustion or of evaporation, J k g fire is critical for fire resistance and fire spread
m mass flow rate, kg/s considerations. It is the postflashover fire that causes the
Q
heat rate, W main fire assault on the compartment boundaries, as well as r stoichiometric air-to-fuel ratio fire spread to neighbouring spaces through the compartmentT temperature, K openings.
Greek Ahhabet Since the pioneer work by Kawagoe (1958), a
significant amount of work has been done on compartment
6 radiative loss parameter = A JA, fires involving cellulosic materials (see a recent review of
E emissivity, dimensionless postflashover compartment fires by Harmathy and Mehaffey 45
(1 983)). Over the past decade, however, a number of studies on compartment fires involving plastic materials have also appeared, as a result of the increasing use of plastics in new furnishings. For example, an experimental and theoretical analysis was carried out on small-scale compartment fires involving plastics by Quintiere et al. (1978) highlighting the dependence of the fire on fuel area and ventilation opening. Small-scale experiments and modeling were also conducted by Bullen and Thomas (1978) showing that thermal radiation is the dominant heat transfer mechanism in fires involving thermoplastic
materials. The effects of charring fuels (typically cellulosics) and non-charring fuels (thermoplastics) on fire
characteristics were discussed by Harrnathy (1 979) who suggested that thermoplastics have a higher potential for fire spread to neighbouring spaces. 4 model that includes the effect of compartment size on combustion efficiency was proposed by Takeda and Akita (1982). Their model relates the combustion efficiency (a measure of the consumption efficiency of the oxygen entering the compartment) to the ratio of the ventilation-air residence time and the turbulent mixing time, and was found to compare favourably with subsequent experiments by Takeda (1 985) using PMMA (polymethylmethacrylate) as fuel and various-sized, small- scale compartments.
In all of these previous studies on compartment fires involving thermoplastic materials, the authors have been concerned principally with the development of the basic equations that govern the fire characteristics. These
equations relate a large number of physical parameters and, often, unknown or difficult-to-determine coefficients such as the combustion efficiency, the radiation configuration factors, the flame and gas emissivities, etc. The values of these coefficients, if they can be determined at all, are usually obtained via lengthy calculation procedures. To provide a simpler mathematical model for conservative fire design considerations, study is needed to see whether the values of these coefficients can be approximated by their limiting values, i.e., maximum or minimum values.
Whether these coefficients can be approximated by their limiting values depends on how sensitive the solutions are to these coefficients. With the large number of physical parameters and coefficients present in these previous studies, it is difficult to examine the sensitivity of these models to the coefficients. One way to examine the sensitivity of a model to its coefficients is to develop the model through similarity considerations, where the fire behaviour can be described through the use of a smaller set of appropriate similarity parameters and corresponding similarity solutions. The smaller set of similarity parameters allows more readily the mapping of the solutions and therefore the determination of the sensitivity of the model to the coefficients to see whether these coefficients can assume their limiting values for a simpler and conservative fire model.
assuming limiting values for the coefficients are then examined to see whether they are suitable for use in conservative fire resistance and fire spread design considerations.
PRESENT MODEL
In the present study, the postflashover compartment fire is considered to be a quasi-steady compartment fire of non-charrina thermo~lastic materials. where the combustible vilatiles arggenerated by the vaporization of the plastic fuel as a result of thermal radiation feedback from the flames and ~-~ ~ ~ - ~
-hot gases and from the compartment boundaries. The quasi-steady-state approximation is justified as the model to be developed is intended to give a general characterization of the burning behaviour of thermoplastic materials, and, for fire resistance and fire spread design considerations, the extreme burning behaviour at the limiting conditions.
The present model also assumes that this is a ventilation-controlled fire, where the rate of combustible volatile generation in the compartment is such that it could support a larger fire than that which could be supported by the fresh air supply. The same modeling approach can be applied to the opposite fuel-controlled fire, where the volatile generation rate is small, relative to the fresh air supply, and limits the fire. Such fuel-controlled fires, however, are less severe than ventilation-controlled fires and are therefore less of a problem.
Also in the present model, the heat loss through the compartment boundaries is neglected. This gives rise to a higher prediction of the compartment temperature and the burning rate as a result of higher thermal feedback. It should be noted that the heat loss through the compartment
boundaries can always be included, as has been done by others, but is neglected to avoid unnecessary complication in a model which is intended to provide a conservative estimate of the fire behaviour for design considerations. In addition, the thermal resistance of compartment boundaries can vary from very low to very high and the present model is simply taking the worst case of a very high thermal
resistance. This approach, similar to the adiabatic approach in thermodynamic analysis, is useful as it simplifies the number of controlling parameters in the problem and gives a conservative estimate of the compartment temperature and burning rate. Both the simplification and the conservative result are desirable in fire prevention design considerations. GOVERNING EQUATIONS
Given the aforementioned assumptions for the present model, the general energy balance for a
postflashover fire involving thermoplastic materials can be written as:
In the present paper, the equations describing the where dynamics of postflashover compartment fires involvina
thermoplastic materials are analyzed to yield the app?oprlate Q, = rate of heat generation by combustion, dimensionless similarity parameters and two coefficients,
Q, = rate of heat absorption by the plastic fuel in and to show that the dependence on these parameters and
coefficients can be characten'zed by two algebraic vaporizing,
equations. The solutions are maD~ed out usina these
6,
= rate of convective heat loss through the similarity parameters and coefficients. ~ o l u t i o h obtained compartment opening,assuming reasonable values for the two coefficients are
Q
= rate of radiative heat loss through the same ~0rfI~ared with available ex~erimental data. Those obtained opening.In equation (I), the heat generation rate by radiative heat transfer to the fuel surface dominates over combustion can be written as: convective heat transfer. The heat absorbed by the plastic
fuel melts and vaporizes the solid fuel. The heat absorption and the net heat exchange between two, non-black,
& =
~ , A H ~ = ~ , ? A H , (2) radiating bodies can be expressed (see Holman, 1981) as:where 4
Q, = me [AH. +
cg
(Te - Tall = qe Ae 0 (Tg -T;I)
(4)m = fuel combustion rate in the gas phase,
AH, = heat of combustion, where
q, = combustion efficiency within the compartment,
ma = mass air flow rate,
r = stoichiometric air-to-fuel ratio.
The combustion efficiency, q,, defined as the ratio of the actual fuel combustion rate in the compartment to the stoichiometric fuel combustion rate in the compartment, measures the efficiency of the consumption of the oxygen entering the compartment. The value of q,, however, cannot be assessed accurately, even though its value normally lies in the range from 0.5 to 1.0 for small-scale to full-scale compartment fires (Takeda and Akita, 1982). In the present model, q, is considered to be one of the two difficult-to- determine coefficients the values of which cannot be exactly determined but can be assumed at its maximum value for conservative design applications. For example, as will be discussed under Parameters and Solutions, q, can be assumed to be 1 (its maximum value) for the worst-case design considerations. The mass air flow rate into the compartment, ma, depends mainly on the opening geometry and, to a lesser extent, on the temperature and pressure fields developed within the compartment (Harmathy, 1980).
For simplicity, and following Harmathy and Mehaffey (1983) and Takeda and Akita (1982), ma in the present model is assumed to depend only on the opening geometry, as expressed in the following relationship:
where
Co = flow coefficient,
@ = ventilation factor, pa = ambient air density,
A, = area of the ventilation opening, g = gravitational constant,
h, = height of the ventilation opening. In the present study, the flow coefficient Co is taken to be 0.1 38, following Harmathy and Mehaffey (1 983) and assuming that the volatilization of the plastic fuel has only a secondary effect on the air flow rate. It is recognized that a large amount of volatilization of the plastic fuel may reduce the air flow rate into the compartment. By neglecting this effect, the present model assumes a slightly higher air flow rate and hence a slightly higher, or conservative,
combustion rate.
In equation (I), the heat absorption rate of the plastic fuel,
Q,,
is governed by the thermal radiation feedback from the flames and hot gases and from the compartment boundaries, and by the re-radiation from the fuel surface. In a postflashover compartment fire where the temperature of the flames and hot gases is very high (1 000 to 1300 K),with
me = fuel evaporation rate or burning rate, AH, = heat of evaporation,
Cg = specific heat of the hot gases, qe = evaporation efficiency,
A, = fuel surface area under thermal radiation,
Ag = equivalent radiating surface of the flames, hot gases and compartment boundaries combined,
F, = effective configuration factor,
ee = emissivity of the fuel surface,
eg = emissivity of the flames and hot gases, o = Stefan-Boltzmann constant,
Tg = temperature of the flames and hot gases, T, = temperature of the fuel surface,
Ta = ambient temperature.
Here, the specific heat of the solid fuel is assumed, for simplicity, to be the same as that of the hot gases. In reality, the specific heat of most thermoplastic solids (1.1 to 1.9 kJ/kg*K (Drysdale, 1985)) is slightly higher than that of the hot combustion gases (about 1.2 kJ/kg*K (Quintiere et al., 1978 and Harmathy, 1979)). Using a slightly lower specific heat for the solids does not significantly affect the heat absorption term because the term includes a larger heat of evaporation component. In any case, a slightly lower specific heat of the solid fuel in the present model allows for a slightly higher, or conservative, burning rate. In
equation (4), the radiating flames and hot gases and the compartment boundaries are treated as if they have been combined into one equivalent radiating surface,
4,
with the same emissivity, Q, and temperature, Tg, as those of the radiating flames and hot gases (in a ventilation-controlled compartment fire of thermoplastic materials, the flames and hot gases are well mixed throughout the wholecompartment). The configuration factor, Fe, is an effective view factor from the fuel surface to this equivalent radiating surface. The evaluation of these effective parameters, Ag
and
F,,
and the gas emissivity, Q, normally a difficult task with great uncertainty in the results, is avoided by combining all these parameters through equation (5) into one, single coefficient, q,, called here the evaporation efficiency. Considering the high emissivity of most thermoplastic materials (Tewarson, 1980), the large configuration factor expected in a confined compartment fire, and the typical dependence of the gas emissivity on the size of the flamesand therefore the size of the compartment (Babrauskas and Williamson, 1978), the value of q, is probably quite high, from 0.8 to 1.0 for small-scale to full-scale compartment fires. In the present model, q, is considered to be the other of the two difficult-to-determine coefficients the values of which cannot be exactly determined but can be assumed at its maximum value for conservative design applications. For example, as in the case of the combustion efficiency, qc, the
evaporation efficiency, lev can be assumed to be 1 (its
maximum value) for the worst-case design considerations.
The fuel evaporation temperature, Te, in equation (4) is
taken to be 673 K, a typical surface temperature of burning solids (Drysdale, 1 985). The measured surface temperature of most burning thermoplastic materials is in the range of 673 to 773 K (Tewarson, 1980). By using a lower surface temperature, which lowers the surface re-radiation, the present model allows for a higher, or conservative, burning rate. The fuel area under radiation,
&,
is well defined if the fuel load is a simple slab, but is not so obvious if the fuel load is complex, such as a crib. The proper area to use for complex geometries is discussed by the author in a separate paper based on his experiments with plastic cribs (Yung, 1 989).The second term on the right side of equation (1) is the convective heat loss term:
Equation (6) simply relates the convective loss to the mass flux through the system multiplied by the change in enthalpy. This equation does not include the enthalpy change when the fuel changes from a solid to a vapour; this was already
considered as the heat absorption term,
Q,.
The last term in equation (1) on the right hand side is the radiative heat loss through the compartment opening, which, following Babrauskas and Williamson (1978), can be simply expressed as:
where the emissivity of the compartment opening is considered to be 1.
These seven equations constitute the basic equations for the problem. They will be solved in the next section.
where the dimensionless parameters are defined as:
3
A, o AH,
AH
a
= ; < = - A ; a = -@
C: AH, Ae&
(10) Equations (8) and (9) show that the two dependentvariables in the problem, the dimensionless temperature,
eg,
and the dimensionless evaporation rate, mJ@, are
dependent on nine dimensionless parameters, six of which are similarity parameters and three of which are coefficients as:
md@ = f2 (r,
c,
9,;ea,
a, 6; CO, 'lc, qe). (1 2)The two dependent variables, €Ig and mJ@, relate the
temperature of the hot gases to the heat of combustion, and the evaporation rate, or burning rate, to the ventilation factor.
Basically they relate the convective heat loss to the heat generated by combustion, and the burning rate to the air ventilation rate. The first three independent similarity
parameters, r,
c
and €I,, are chemical and thermodynamicproperty parameters that define the stoichiometric air-to-fuel ratio, the ratio of heats of combustion and of evaporation, and the surface evaporation temperature. The next three
independent similarity parameters,
ea,
a, and 6, areparameters that define the physical conditions of the
problem. The dimensionless temperature,
ea,
defines theambient temperature. The radiation area parameter, a, relates the radiation area to the ventilation factor. Since the radiation area affects the heat absorption by the fuel, and the ventilation factor affects the combustion rate, a basically measures the amount of heat absorbed by the fuel relative to that generated by combustion. The radiative loss parameter, 6, measures the heat radiated through the compartment opening relative to that absorbed by the fuel. Basically, a
PARAMETERS AND SOLUTIONS and 6 are the only two similarity parameters that one can
have some control of. The three coefficients, C,, qc, and qe,
Based on similarity considerations and the are not, strictly speaking, independent parameters but are
appropriate similarity Parameters to use, equations (1) to (7) dependent on other physical parameters of the problem.
can be combined and re-arranged into the following two However, their values are roughly known and can be
non-dimensional equations that basically characterize the considered independent parameters with certain assumed
problem: values, as have been discussed earlier.
The two dependent variables, €Ig and mJ@, are
governed by equations (8) and (9). Equation (8) is a 5th degree algebraic equation for Bg, which cannot be solved to
C ( e a - e e ) express
eg
as an explicit function of the other independent1 +
5
(ee-
9,) parameters. However, equation (8) shows the independentparameter a as an explicit function of Og and the other
PRESENT THEORY
---
qc=qe=l.o-
---
EXPERIMENT-
6 -1.1 -1.7 0.16 PRESENT THEORY qC= 9 = 1.0 (THEORETICAL MAXIMUM) qc =0.58. q,=0.84----
0.10Fig. 1 For PMMA, dimensionless gas temperature (Og) as a function Fig. 2 For PMMA, dimensionless burning rate (m&) as a function of the radiation area parameter (a) for various values of the of the radiation area parameter (a) for various values of the radiative loss parameter (6) and two different values of radiative loss parameter (6) and two different values of combustion efficiency (qc) and evaporation efficiency (qe) combustion efficiency (qc) and evaporation efficiency (qe)
be obtained by plotting a as a function of eg, and then inverting to show Bg as a function of a. The results from equation (8) can be substituted into equation (9) to obtain the solutions for the other dependent variable, m$$, which
can also be plotted with m& as a lunction of a and the other independent parameters, This numerical procedure Is used for the case of a comparlment fire involving
PMMA
(a thermoplastic fuel), and the predictions are plotted in Figs. 1and 2. In these figures, the dimensionless gas temperature,
eg, and the dimensionless burning rate, mJ$, are plotted as
functions of the radiation area parameter, a, for various values of the radiative loss parameter, 6, and for two sets of values of the combustion efficiency, qc, and the evaporation efficiency, qe. For PMMA, the stoichiometric air-to-fuel ratio, r, is 8.25 and the heats ratio, <, is 25.7 (Quintiere et al., 1978 and Harmathy, 1979). The dimensionless evaporation temperature, Oe, is 0.031 1 and the dimensionless ambient temperature, ea, is 0.01 38, based on an evaporation temperature of 673 K, an ambient temperature of 298 K, a specific heat of 1.20 x 103 J/kg*K, and a heat of combustion
of 26.0 x 106 Jlkg. The flow coefficient, Co, has a value of 0.1 38 as was discussed earlier.
The results in Figs. 1 and 2 show that when the radiation area parameter, a, is small, the dimensionless gas temperature, eg, drops and the dimensionless burning rate,
m$$, increases with an increase of a. This is physically
reasonable as a is a measure of the relative heat absorbed by the fuel, and
eg
is a measure of the relative convective heat loss. An increase in a should reduce the convective heat loss and should increase the burning rate. Figures 1 and 2 also show that when a is large, m$$ flattens or evendecreases. This is again physically reasonable because
when a is large, Og is quite low and the burning rate should slow down as a result of lower thermal radiation feedback. The figures also show that both
eg
and m,J$ decrease invalue if the value for the radiative loss parameter, 6, is increased. Since 6 is a measure of the relative radiative heat loss through the compartment opening, an increase in 6 should reduce both the relative convective heat loss and the burning rate. Also shown in the two figures are that both
eg
and m$$ increase with an increase in the combustionefficiency, qc, and the evaporation efficiency, qe. Not as obvious (but easily proven by examining equations (8) and (9)) is that an increase in qc increases both Og and mJ$,
whereas an increase in qe increases mJ$ but decreases g.I€
This is again reasonable as an increase in qc would
increase the heat generation and hence both the convective heat loss and the burning rate. An increase in qe, on the other hand, would increase the heat absorption rate, resulting in an increase in the burning rate and a reduction in the relative convective heat loss. For qc = q, = 1, the burning rate is the theoretical maximum, whrch is seen to be about twice the nominal values. The theoretical maximum, therefore, is not an unreasonable upper limit to be used in worst-case dssign considerations to provide a reasonable margin of safety. In addition, as was discussed earlier, both the combustion efficiency, q,, and the evaporation efficiency,
qe, depend on compartment size and their values could be quite close to 1 in full-scale compartment fires. The theoretical maximum for the gas temperature is harder to define, since it requires that the evaporation efficiency be zero, a physically unrealistic solution. But since for
qc = qe = 1, the gas temperature is relatively high, it can be considered as a practical maximum, especially at lower values of a.
In the above calculations, the ventilation control condition requires that the fuel evaporation rate, me, be higher than the fuel combustion rate, m,, or higher than q c ma/'. This condition can be expressed by the following inequality:
where the burning rate must be greater than q,Cdr, the combustion limit. The difference between the burning rate and the combustion limit is the rate that the unburned fuel is being convected to the outside through the compartment opening. This difference is important for flame projection studies and can be obtained from Fig. 2. In flame projection design considerations, the maximum curve (q, = qe = 1) should be applied. The theoretical maximum burning rate, as was discussed earlier, provides a reasonable safety factor for design considerations. In fire resistance studies, the compartment temperature and the fire duration are critical. The maximum gas temperature and the maximum fire duration can be obtained by using the maximum curve for the gas temperature and the maximum combustion limit as the slowest burning rate. Looking at Fig. 2, the slowest burning rate based on the maximum combustion limit can be quite low when compared with the actual burning rates, especially with those of full-scale compartment fires where the burning rates may be quite close to the maximum curve. However, as was mentioned earlier in the paper, there is no accurate way of predicting the burning rate even with more complex models because of the difficulties in estimating the values of some of the controlling parameters such as the combustion efficiency, the radiation configuration factors, the flame and gas emissivities, etc. The present model at least gives the slowest burning rate and the maximum burning rate (from the maximum curve) and hence the upper and lower bounds of the fire duration. Until such time when there are more experimental data available to define a more reasonable fire duration for fire resistance considerations, the maximum fire duration should be used to provide a certain margin of safety. In any case, as will be seen later in this section, the maximum fire duration based on the slowest burning rate is not such an extreme because it is quite close to that of cellulosic materials.
For comparison, small-scale experimental data by Quintiere et al. (1978) and Takeda (1985) in the ventilation controlled regime are also plotted in Figs. 1 and 2. Since the ventilation opening area was not explicitly given in the data by Takeda, the radiative loss parameter, 6, is assumed to be approximately 1, based on the observation that the opening area and the fuel area are about the same. The comparison shows, as expected, the theoretical maximum gas temperature and the theoretical maximum burning rate as the upper bounds of the experimental data. The
maximum gas temperature is seen to be slightly higher than that of the experiment, and the maximum burning rate is about twice that of the experiment. It should be noted, as was discussed earlier in the paper, the burning rate of a full- scale compartment fire could be higher than that of a small- scale compartment fire, and the difference between the theoretical maximum and the experiment could be much closer. The comparison also shows that the theory and the experiment agree quite well, based on a reasonable estimate of the combustion efficiency (0.58) and the
evaporation efficiency (0.84). The values assumed for the two efficiencies are not unreasonable for small-scale compartment fires (the probable values of these two difficult- to-determine coefficients were discussed earlier under Governing Equations). The model also compares favourably with data (not shown) obtained by the author using PMMA cribs (Yung, 1989). Although the comparison in Figs. 1 and 2 is made using assumed values for the two efficiencies, the model is still validated because of the fact that it compares favourably with N data points using only two assumed values (not two data points using two assumed values which is always possible). For design purposes, the exact knowledge of these two efficiencies is not critical; maximum values are recommended in order to provide a certain margin of safety.
For reference, the burning rate for cellulosics under ventilation control (Harmathy, 1979) is also plotted in Fig. 2. The plot shows that the maximum combustion limit of the PMMA is not very different from the burning rate of the cellulosics, and that most of the plastic fuel vaporized is not burned, but vented outside of the compartment. For fire resistance design considerations, therefore, the maximum fire duration for thermoplastic materials, based on the maximum combustion limit as the slowest burning rate, is quite comparable to the fire duration of cellulosic materials.
EXAMPLE
For simplicity, assume there is a regular-size,
compartment fire with PMMA as fuel. Assume the ventilation opening (for example, a door) is I m wide by 2 m high, and the PMMA has a surface area of 10 m2 and a total mass of
1190 kg. Based on the definitions given in equations (3) and (1 O), the ventilation factor, @, is 10.6 kgls, the radiation area parameter, a, is 0.0265, and the radiative loss parameter, 6, is 0.2. From Figs. ( I ) and (2) and using the limiting solutions (qc = qe = I), the dimensionless gas temperature,
eg,
is 0.061 1, the dimensionless burning rate, mJ@, is 0.1038, and the combustion limit is 0.0167. For exterior fire spread considerations, the maximum venting of unburned fuel is the difference between mJ$ and the combustion limit. The difference is 0.0871 which gives a mass venting rate of 0.926 kgls and a possible heat release rate from external combustion of 24.1 MW. Inside the compartment, the combustion rate can be obtained by definition from the combustion limit, which gives a mass combustion rate of 0.1 77 kgls and a heat release rate of 4.6 MW. The gas temperature, fromeg,
is 1320 K. For compartment fire resistance considerations, the slowest burning rate (i.e., the combustion rate and not the evaporation rate) should be used to calculate the fire duration. Based on the slowest burning rate of 0.1 77 kgls, the fire duration is 1.87 hr.CONCLUSIONS
In this paper, the equations that govern postflashover fires of thermoplastic materials were analyzed through similarity considerations. It was found that by neglecting the heat loss through the compartment boundaries, the
equations can be combined into two algebraic equations to give conservative estimates of the compartment temperature and the burning rate. The two dependent variables (the dimensionless gas temperature and the dimensionless
burning rate) were found to depend on six similarity parameters and three coefficients. The first four
independent similarity parameters are property and ambient condition parameters: the stoichiometric air-to-fuel ratio, the ratio of the heats of combustion and of evaporation, and the dimensionless evaporation and ambient temperatures. The problem therefore depends on only two controllable similarity parameters. They are the radiation area parameter, which measures the relative amount of heat absorbed by the fuel in evaporating, and the radiative loss parameter, which measures the relative amount of radiative
, heat loss through the compartment opening. The three
coefficients are the flow coefficient, the combustion efficiency and the evaporation efficiency. The value for the flow coefficient is well known, whereas the values for the two efficiencies are not well defined, except that they are known to be less than 1 and depend on the compartment size. The solutions were mapped out with the two dependent similarity parameters as functions of the two controllable similarity parameters and for different values of the two efficiencies. For conservative fire design considerations, the limiting solutions with the two efficiencies equal to 1 are
recommended. Comparisons with available experiments showed good agreement between the present model and experimental values. The present model provides a simple tool for use in fire resistance and fire spread design considerations. The present model can also be used as an analytical reference for checking complex numerical solutions at the limiting conditions.
ACKNOWLEDGEMENTS
The author would like to thank J. K. Richardson, Section Head, for his encouragement to publish this paper. REFERENCES
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