Probabilistic models for cable tray fires

Reliability Engineering3

(1982) 2 1 3 - 2 2 7

PROBABILISTIC

MODELS

FOR CABLE TRAY FIRES

N. O. SIU • G. E. APOSTOLAKIS

School of Engineering and Applied Science, University of California, Los Angeles, California 90024, USA (Received: 20 July, 1981)

ABSTRACT

A probabilistic modelfor determining the time-dependent severity ofa nuclearpower plant compartment fire is proposed. The model is developed by analysing the frequency of the loss of two electrical divisions in the cable spreading room. The methodology requires the use of a physical modelfor fire, analysis of the uncertainties in this model due to statistical and state-of-knowledge uncertainties, a model for fire suppression and a model for the failure of cables due to thermal effects.

NOTATION

Area (m 2) Specific heat (J/kg K) E~ Error factor for zc (see eqn. 9) Fo_s Shape factor from source to object Conditional frequency of fire spread, given a fire (eqn. 12) L Heating value of fuel (J/kg) h Heat transfer coefficient (W/m 2 K) Mass of fuel (kg) • ii Specific burning rate constant (kg/m 2 s) mo Heat content of pilot fuel (Btu) Qp q; Heat flux to object (W/m 2) T Temperature (K) Spontaneous ignition temperature (K) (v) Average flame front velocity (cm/s) 213 A

Cp

Reliability Engineering0143-8174/82/0003-0213/$02.75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain

214

0 TG TS

0

N . O . SIU, G. E. APOSTOLAKIS

Thermal diffusivity (m2/s) Emissivity Dimensionless ratio of characteristic times (eqn. 6) Stefan-Boltzmann constant (5.6697 × 10-s W/m 2 K 4) Mean fire growth time (min) Mean fire suppression time (min) Dimensionless variable (eqn. 8)

Selected subscripts e f o p s

Environment Fuel Object Pilot Source

1.

INTRODUCTION

One of the key elements in the quantification of the fire risk in nuclear power plant is the estimation of the frequency of component damage caused by fire. This estimation process in turn requires models for the frequency of fire occurrence within a power plant, the distribution of the magnitude of fire severity given that a fire occurs and the frequency of component failure upon exposure to fire of a given severity. In this work, we concentrate on developing the second model of these three, i.e. the probabilistic model for the time-dependent thermal hazard a component is exposed to during a nuclear power plant compartment fire. The frequency of these compartment fires has been assessed by Apostolakis and Kazarians, 1 and simple component fragility models can be adopted in the analysis. Our approach requires the construction of a physical model for the fire scenario of interest. This model, outlined by Siu 2 (see this issue), is called the deterministic reference model (DRM). The uncertainties in the input parameters used by the DRM are propagated through the DRM and are then combined with a probability distribution quantifying our uncertainty in the accuracy of the DRM, to form a probability distribution for the frequency that a component is exposed to a specified thermal hazard (e.g. high temperature). Relevant data to aid our construction of the parameter and modelling uncertainty distributions are often unavailable and so subjective judgement is an important input in this process. As a vehicle to illustrate our approach, we construct a distribution for the frequency of the loss of cables in two electrical divisions in the cable spreading room, given that there is a fire in that room. Models are incorporated for the competing processes of fire growth and suppression, and an extremely simple and conservative cable failure criterion is employed.

215

PROBABILISTIC MODELS FOR CABLE TRAY FIRES

2.

PROBLEM DEFINITION

The physical configuration of the fire zone to be analysed is shown in Fig. 1, each horizontal row of cable trays representing one electrical division. The trays are 5 ft long, and are completely filled with cables. There are no thermal barriers between or within trays and no flame retardant coatings have been applied to the cables. Preliminary calculations indicate that a fire on an upper level tray will not propagate to a lower tray, if the fuel bed geometry is constant. In the case of a

® I

® I L

I 4fit

©

I

II

I_ I~

Fig. 1.

12 IN.

_1

- I

Cable tray configuration.

massive fire, the upper tray may collapse, but we do not analyse this situation, because the same massive fire will already have caused the loss of two or more divisions. Since the lower trays are more accessible, and hence have a greater exposure to transient fuels and ignition sources, we assume that the initial fire is established on a small portion of tray 1, as seen in Fig. 1. Our task is to determine the likelihood that any trays in the level immediately above are damaged by the fire starting in tray number 1. Clearly, tray 2 is the most vulnerable tray, but our approach can easily account for the other trays as well. Cable insulation thermal damage thresholds are typically given in terms of a critical temperature level, above which the insulation begins to deteriorate. Achievement of this temperature at the cable surface does not necessarily lead to immediate loss of electrical circuit integrity; the temperature profile within the insulation generally drops with distance from the surface, and thus the deterioration wave will require some amount of time to progress to the actual wire. To simplify our modelling, we assume that the cable is lost if the insulation ignites, the ignition temperature generally being somewhat higher than the deterioration temperature. We further treat each unit length of cable tray contents as a single fuel element, i.e. all cables in a single cross-section are exposed to the same amount of heat and have the same temperature. Thus, if ignition is achieved at one location on a tray, all cables passing through this tray are assumed lost. The result of this conservative modelling of cable failure is that the main thrust of our modelling effort can be directed towards determining the frequency of fire spread from tray 1 to the neighbouring trays.

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N. O. SIU, G. E. APOSTOLAKIS

3.

THE FREQUENCY OF FIRE SPREAD

The frequency of fire spread to neighbouring trays is a function of the characteristic times of fire growth and suppression, 36 and r s, respectively. If 36 is less than rs, the fire spreads to tray 2, tray 3, or any other tray, depending on how 36 is defined. We model 36 and zs independently, primarily because of the crude suppression model utilized. More detailed efforts should certainly account for the dependence of zs on the size and intensity of the growing fire.

3.1. Fire growth modelling As in the case of cable deterioration, cable ignition occurs when the insulation surface reaches a critical temperature. The time to reach this temperature is a function of the heat flux exchange at the cable's surface between the cable and the external environment and of the physical characteristics of the cable itself. The two primary modes of heat transfer at the surface are convection and radiation. For convection, the net heat flux into the cable is proportional to the temperature difference between the immediate environment and the cable surface, i.e.

3'0'=h[T(7, t ) - To]

(W/m 2)

(1)

This mode is important when the cable is immersed in the flames or in the hot buoyant plume above the flames. For radiation, the net heat flux is proportional to the difference of the radiation source temperature and the cable's surface temperature raised to the fourth power: (10 = ~aFo-~[T~ - To4]

(W/m2)

(2)

The shape factor Fo_ ~ represents the geometrical attenuation of the heat flux in travelling from source to the cable. Note that we frequently write the term a T ) as q~', and do not explicitly use the source temperature Ts in our calculations. The contribution of thermal radiation to the total input heat flux at the cable's surface generally dominates the contribution from convection, except in the zone of hot gases within and above the flames, where the contributions are of the same order. The models required to compute q~' and F o_S are quite simple when the source fire involves a fixed pool of fuel which is far removed from any other combustible objects. The calculations for 4~' and F o_2 are then time-independent, since only the ignition of the target cable will cause a change in the fire size. Basically, the flame height Zfl and the specific mass burning rate rn" are computed using appropriate correlations; the former is used in conjunction with the pool diameter and the distance to the cable to find Fo_ s, and the latter, when multiplied by the pool area and the heat output per unit fuel burned and divided by the flame surface area, is proportional to q~'. When the starting fire involves only a portion of the initial fuel

PROBABILISTIC MODELS FOR CABLE TRAY FIRES

217

bed or is nearby other fuel elements, the growing fire requires time-dependent calculations which are algebraically more complex. Both the static and the dynamic calculations can be done using the simple model described by Siu 2 and implemented using the computer code C O M P B R N . 2 The model incorporates many of the important physical aspects of fire, and is shown to make reasonable predictions when used to simulate a number of experimental fires. However, the model is deterministic; its predictions by themselves do not indicate the associated large uncertainties. For example, if specific parameter values are used as input to C O M P B R N , a single estimate for z G will result. Yet we are not sure that these parameter values are accurate, nor even that COM PBRN will yield the correct value given the correct input. The physical bases for these uncertainties are also described by Siu; 2 we now demonstrate how they are to be quantified in our analysis.

3.2. Classification of uncertainties The uncertainties in our prediction of the fire growth time r~ can be classified as statistical uncertainties and state-of-knowledge uncertainties. 3'4 The statistical uncertainties stem from the random nature of fire; if we perform a particular experiment a large number of times under 'identical' conditions and measure r~, we will obtain a frequency distribution. Our physical model cannot predict the environmental fluctuations which lead to this distribution. Rather, it utilizes timeand spatially-averaged quantities in deriving the estimates of heat generation and absorption, and eventually fire spread. Thus, we henceforth regard the physical model's prediction of z G as an appropriate measure of central tendency (e.g. mean or median) for the frequency distribution of the spread time. The exact definition of z~ is not important, since our state-of-knowledge uncertainties are felt to be considerably larger than the' statistical uncertainties. The statistical uncertainties described above are inherent in the nature of fire. Even if our knowledge of fire were to increase dramatically, we would not be able to reduce these uncertainties significantly. Such an increase in knowledge, however, would markedly reduce our state-of-knowledge uncertainties. These uncertainties arise when we attempt to predict the frequency distribution for a parameter of interest, applying a 'simple' model (simple in the sense that it is solvable) to a complex problem.5 As discussed in ref. 2, major uncertainties in the modelling of the given fire arise from uncertainties in the fundamental physical modelling and in the values of the parameters to be used when the model is applied toward a real compartment fire. To illustrate these different sources of uncertainty, we consider the growth period model presented in ref. 2. The growth period model is a synthesized model; its components are the many models available in the literature which describe various physical phenomena, including models for burning rate as a function of ventilation, flame height as a

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N. O. SIU, G. E. APOSTOLAKIS

function of burning rate, radiated heat flux as a function of flame height, etc. In general, each of these models was developed independently. Our uncertainty in the modelling of the basic physics of fire, therefore, is due not only to our uncertainty in the accuracy of each model's predictions under the conditions they were developed for, but also to our uncertainty in the synthesis of these independent models. One of our primary concerns is whether or not the synthesis contains enough component models (i.e. if all important phenomena have been modelled). As for the parameter uncertainties, we discuss in ref. 2 some of the uncertainties in the physical properties of cable insulation which strongly affect our confidence in the fire spread calculations. These uncertainties are associated with a particular cable type; since there actually is an unknown mix of cable types within the cable spreading room, and even nominally identical cables from different batches may exhibit some variations, the uncertainties in our calculations resulting from the use of fixed physical property values can be important.

3.3. Quantification of uncertainties The probability distributions for re, where we recall that r C is an 'average' for the growth time frequency distribution, are constructed using three elements: a deterministic reference model (DRM), a set of probability distributions for each input parameter required by the DRM and a probability distribution for a parameter which measures our confidence in the accuracy of the DRM. 3.3.1. The deterministic reference model. The DRM for our problem is the fire growth model presented in ref. 2; it explicitly incorporates the physics of the problem into the analysis. Given a set of input parameters, the DRM yields point estimates for z~, which we treat as an 'expert's opinion'. When we later quantify the uncertainties in the DRM, we are expressing our belief in the credibility of the expert in a spirit similar to the approach used by Apostolakis and Mosleh. 6 It should be noted that the DRM does not necessarily provide a 'best estimate' output, although efforts are usually made to ensure that it does; if there are competing models available, simplicity and ease of use are also important factors in the choice of the reference model. Some comments pertaining to our reference modelling of the cable tray fire are considered: (1) (2)

(3)

The cable trays are 5 ft long, and are divided into five discrete fuel cells, each being 1 ft square. The fire starts on the middle fuel cell of tray 1, and additional piloting fuel (e.g. oil or trash) may be present in this cell. The pilot fuel is discussed in the next section. The remaining fuel is cable insulation. The porosity factor, the factor which modifies the fuel cell area to account for the surface roughness which increases the actual area available for burning, is set equal to 3.14.

PROBABILISTIC MODELS FOR CABLE TRAY FIRES

(4)

(5)

219

No credit is given to the shielding effect from the bottom of tray 2 or any other trays; the flames on tray 1 are assumed to transmit heat directly to the fuel in tray 2 with no attenuation by the intervening material. This, of course, is a conservative assumption. Enclosure effects are neglected. The fire is considered to be too small to cause a significant hot gas layer near the ceiling. We note that this neglect may be non-conservative when the trays are near the room walls, since reflection from the walls may be important.

3.3.2. Pr opagation of parameter uncertainties. OncetheDRMisconstructed, the uncertainties in the input parameters are propagated through the DRM. The parameters which are varied in this simulation are the specific burning rate constant rho, the cable insulation heating value Hf, the spontaneous ignition temperature T* and the pilot fuel heat content Qp. The first three parameters are discussed in ref. 2; the pilot fuel variation represents our uncertainty in the initial conditions of the fire and requires further discussion. As has been noted previously, cable insulation is generally a difficult material to burn. The type and amount of pilot fuel involved in the initial fire is thus an important factor in determining the rate of fire growth. If there is a large amount of oil, rapid growth is nearly assured. If negligible amounts of additional fuel are present, fire growth may not occur at all. We model the pilot fuels with a single parameter Qp, the amount of heat released when the pilot fuel is consumed: Qp = HrpM p

(3)

The use of this single parameter, while crude, is not entirely unreasonable, since the ignition of a fuel element is governed by the total amount of heat absorbed. A more sophisticated analysis may take the different burning rates of the pilot fuels into account, since the more intense fires, e.g. oil fires, have high flames and thus transfer more heat to the target cable than fires involving less intensely burning fuels. Before we propagate the uncertainties in the four input parameters through the DRM, we choose to construct a response surfacefl using the DRM, which is applicable over the possible ranges of the parameters. This response surface is a simple approximate functional relationship between the parameters and the output variable, zo- Its purpose is to reduce the computational costs incurred when the uncertainty propagation is performed. Construction of the response surface is easily accomplished using the Latin Hypercube Sampling technique, described in ref. 8. This technique requires that maximum and minimum bounds be assessed for each input parameter. Combinations of the input parameters are then generated in such a way that the full range of each parameter is represented in the DRM calculations. The parameter combinations, or input vectors, are processed by the DRM, and a simple function is

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N. O. SIU, G. E. A P O S T O L A K I S

fitted to the resulting data points using some suitable procedure (e.g. least-squares regression). The bounds for m~, Hf and T* are given in Table 1. We note that two sets of trials were used to construct the response surface. The second set of trials used tighter bounds on the parameters than the first, in order to represent more accurately the cable insulation properties actually expected. The bounds for m o are subjectively derived after consideration of the burning rates of more flammable materials such as TABLE 1 SAMPLING BOUNDS FOR tho, Hi-, T*

Trial set

Parameter

1 1 1

th 0 ( k g / m 2 s) Hf(J/kg) T* (K)

2 2

rh o Hf

2

T*

Lower bound

Upper bound

0.0 1.85 x 107 710 1 x 10 5 1"85 x 107 800

0.0075 2.70 x 107 850

0"003 2"70 x 10 ~ 850

Hf

wood, while the bounds for and T* result from a synthesis of data for plastics and data from ref. 9. The pilot fuel heat content Qp was divided into four discrete classes: 400 Btu, 2000 Btu, 10 000 Btu and 40 000 Btu. To provide some physical basis for the choice of these classes, we note that 400 Btu from a 1 ft 2 fuel packet roughly corresponds to cable insulation burning for one minute (i.e. no additional pilot fuel), 2000 Btu roughly corresponds to wood chips or paper, 10 000 Btu can be associated with oily rags and 40 000 Btu roughly corresponds to a quantity of oil (approximately one quart (1-14 litres)) burning for two minutes. To correlate the C O M P B R N calculations, we can define a characteristic time before self-extinguishment Le, as = (_M[__~

(s)

(4)

~#,hoA f )

"[se

and a characteristic ignition time z~ as (see ref. 2) =

J

_

(s)

(5)

where the denominator within the brackets represents a radiation heat flux to the target cable, when the shape factor from the cable to the flame is 1/7. The ratio of these two times yields a characteristic parameter 0, which upon substitution for constant values, is given by 0

106[.T* -_29812Hf _]

3'96rhox

(6)

PROBABILISTIC MODELS FOR CABLE TRAY FIRES

221

The results of our calculations then indicate that zc is fairly well correlated by re = A(Qp) + B(Qp)O

(7)

where A and B are given in Table 2. The only difficulty with eqn. (7) is that occasionally the DRM-predicted value of zo becomes infinite, even for relatively moderate values of 0. The physical situation is that the flames on tray 1 are moving to the ends of the tray as the fuel in the centre TABLE 2 RESPONSE SURFACECOEFFICIENTS

Qp ( Btu)

A (rain)

B (rain)

400 2000 10000 20 000

0"3 1-1 1.5 2"0

19'8 14"7 8'0 0"0

of the tray is completely consumed. Occasionally, the middle fuel element in tray 2 (the portion closest to the initial fire location and the recipient of the greatest amount of heat) has not absorbed enough heat to ignite before the flames on tray 1 leave the centre of that tray. As a result, the fire on tray 1 exhausts all its fuel and selfextinguishes before tray 2 can ignite. We define a further parameter q;: cp( T* - Te) yHf

= 2610 T~* - 298 Hf

(8)

and our response surface is eqn. (7) with the following condition: zo ~ ~ with a frequency of 87.5 ~o when Qp=400Btu,

0>0.9

and

ff>0.06, or

Qp = 2000 Btu,

0 > 1.9

and

~O> 0.06

It must be emphasized that this response surface is applicable only for the given problem; if the problem geometry is altered, the response surface coefficients will also change. In order to propagate our state-of-knowledge uncertainties in rho, Hf, T* and Qp through the response surface, we must construct distributions for these parameters.

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N . O . SIU, G. E. APOSTOLAKIS

We use the following subjective distributions: m• 0t! = lognormally distributed with # = - 6.91, cr = 0"865 (median = 0.001 kg/mZs, error factor = 4) H e = uniformly distributed between 1.85 x 107 J/kg and 2.7 × 107 J/kg Ts* = uniformly distributed between 800 K and 850 K and Pr(Qp = 400 Btu) = 0.10 Pr(Qp = 2000 Btu) = 0.44 Pr(Qp = 10000 Btu) = 0.44 Pr(Qo = 40 000 Btu) = 0.02 The histogram for Qp should be interpreted as follows. If we examine a large number of fires involving the configuration studied, we will reveal a frequency distribution for the heat released by the pilot fuel. We consider Qp to be a characteristic parameter (e.g. the mean) of this frequency distribution and our uncertainty in the pilot fuel is modelled by our uncertainty in this characteristic parameter of the unknown frequency distribution. Thus, our state-of-knowledge histogram says that we are 10~o confident that the characteristic parameter is 400 Btu, 44 ~o confident that it is 2000 Btu, etc. Based on the above distributions, the probability histogram for r~ reflecting our uncertainties due to parameter uncertainties is shown in Fig. 2. This is a state-ofknowledge distribution for the variable TG.DRM. The large tail above 120min is

.20

Y• o

.lo

20

40

60

80

loo

12o

14o

~'DRM {mln /

Fig. 2.

Distribution of growth time, showing parameter uncertainties only.

160

PROBAB1L1STIC MODELS FOR CABLE TRAY FIRES

223

mainly due to our model's prediction of an infinite propagation time for certain values of 0. We choose to conservatively incorporate this information using a smooth, finite tail, noting that either treatment (i.e. incorporation or neglect) will make little difference when suppression effects are included in the analysis. 3.3.3. Quantifying uncertainty in DRM modelling. The last remaining type of modelling uncertainty to be quantified is the uncertainty in the basic physical modelling of the DRM. We recall that we wish to construct a probability distribution for the average spread time zc, and that we have derived a probability distribution for TG.DRMwhich incorporates our uncertainties in the problem parameters. We now define an error factor E, such that "rG = ErZG,DR M (9) E¢ may be thought of as a factor which measures our confidence in the prediction of the DRM. A distribution weighted to the right of the point E, equals unity indicates that we believe our DRM tends to underpredict the actual value of r G, while a distribution weighted to the left indicates the opposite. We now construct our distribution for E¢ from both the qualitative information presented in Part II concerning the different sources of uncertainty in the DRM and from some rough comparisons of predicted flame spread velocities with published values. There are little data available currently for the fire spread times between noncontiguous cable trays. However, we do possess information for the average flame spread velocity (v) over a single cable tray. If we assume that (/'))experiment

"['DRM

DRM

rexp

1 E,

(10)

we can use predictions of the average flame spread velocity for two simulations to indicate the magnitude of E,. In its modelling of Przybyla and Christian's vertical cable tray experiment, ref. 2 shows that 0"58 ( (1))DRM < 1-27cm/s where 0"08 < (/))exp < 0.36cm/s

Thus, 1-6 < E, < 16. The conservatism of the DRM's prediction is expected for the vertical cable tray configuration, since fuel melting and dripping are ignored. For horizontal tray fires, Pinkel quotes an 'average propagation velocity' of 1 in/min, or 0.04cm/s. 1° Calculation of the flame front velocity along tray 1 in our simulations, where no additional pilot fuel is added, shows that 0-02

<

(/))DRM < 0.14cm/s

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N.O.

S I U , G . E. A P O S T O L A K I S

where the variability is due to parameter uncertainties. The mean of this average velocity distribution is 0.06 cm/s, and the median is 0.04 cm/s. If we use Pinkel's estimate as a reference point: 0.5
3.4. Combined probability distribution jor growth time The average time required for fire spread from the initially burning tray to the tray immediately above, r e, is given by ~CG = ErlTG,DR M

(

1 1)

The first factor on the right-hand side is an error factor representing our modelling uncertainty, while the second factor is a D R M prediction which also incorporates the contribution from input parameter uncertainties. The probability distributions for both factors can be combined, after discretizing the distribution for E~ and using the discrete probability distribution arithmetic described by Kaplan. 11 TABLE 3 HISTOGRAM FOR VERTICAL SPREAD TIME

zc, (min)

Pr ('r~)

10 30 50 70 90 110 > 120

0.26 0-21 0-12 0-08 0.04 0-03 0.26

A condensed histogram for rc is given in Table 3. We note that a significant fraction of fires takes longer than two hours to propagate to tray 2.

3.5. Fire spread frequency including suppression The distribution for rG derived in the last section is a probability distribution for the event 'fire spreads to second division in t min, given an initial fire on tray 1 and ignoring suppression efforts'. In order to compute the conditional frequency that two cable trays, each in a different division, are involved in a fire (given a fire in the cable spreading room), we must include a model for fire suppression.

P R O B A B I L I S T I C M O D E L S F O R CABLE T R A Y FIRES

225

Examination of the data reported by Fleming et a1.12 leads to the following distribution for Zs, the mean time between fire ignition and suppression. We note that the suppression times for fires occurring during plant construction have not been included in the distribution. Pr(z s = 5 min) = 0.40 Pr(r s = 15 min) = 0.30 Pr(zs = 30 min) = 0.20 Pr(% = 60 min) = 0.10 The justification for this distribution is as follows. The estimates reported in ref. 12 are the results of expert evaluation. There are 17 fires that are studied. The experts estimated that seven ( ~ 40 %) of them were extinguished within 5 min. Four of the 17 fires were estimated to have been extinguished between 5 and 25 min after initiation, and another group of four had extinguishment times between 30 and 35 min. Finally, there were two fires that were extinguished after 60 min (but before 85 min). Of course, these estimates are not statistical data, and we really do not know what the experts had in mind when they developed them. We judge that our discrete model is fairly consistent with the expert's estimates. Furthermore, it conforms with our belief that it is very likely that most of the fires in the cable spreading room will be extinguished fairly quickly by the personnel who started them. Let us denote the conditional frequency of fire spread to a second tray, given a fire, by J~. Frequency J'~ is given by f¢ = exp ( - rG/ZS)

(12)

where we have assumed that the time to suppression is exponentially distributed TABLE 4 HISTOGRAM FOR THE CONDITIONAL FREQUENCY OF FIRE SPREAD TO AN ADJACENT TRAY, GIVEN A FIRE Pr(f~) 3.4 3'8 4"3 4-9 5.5 6-2 7-0 4"4 1-4 3"0 5"1 6"9

x 10 - 9

x 10 -8 x 10 -7 x 10 - 6

x 10 - s x 10 - 4 x 10 -3

x 10 -2 x 10-1 x 10 -1 x 10 -1 × 10 -1

0"16 0"02 0"03 0"03 0"06 0"08 0"12 0"10 0"10 0"10 0'10 0"10

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Y. O. SIU, G. E. A P O S T O L A K I S

with mean r s. We note that the frequency if fire spread by time t is also the frequency of non-suppression by that time. The distribution offc is given by P(Jc) = P(r~)P(rs)

(13)

where the distributions of r~ and r s were given previously. It should be noted that in this simple analysis we ignore any dependence between rcJ and rs, although one certainly expects a correlation between the suppression time and the size of the fire (a large fire implying a large rs). The distribution for./~ obtained is listed in Table 4.

4.

CONCLUSIONS

A methodology for utilizing physical models for fire in nuclear power plant fire risk analysis has been proposed. This methodology uses the physical model to provide deterministic reference calculations, upon which statistical and state-of-knowledge uncertainty distributions are then superimposed. The approach is used to determine the distribution of the frequency of loss of two cable trays in a hypothetical cable spreading room due to fire. The width of the resulting distribution indicates the large uncertainties in our current fire-modelling ability. A less obvious but equally important result of our example is that the assessment of the relative likelihoods of the various pilot fuels plays a crucial role in the final results, since large pilot fires almost assure fire spread to the tray immediately above. Refinements of the proposed model are certainly possible and desirable. For example, the analysts may wish to account for the inhibition of fire growth caused by suppression efforts. Agents such as Halon, CO 2 and water can markedly slow down the rate of growth. Indeed, the flame front speed during much of the Browns Ferry fire was relatively low. 13 Rough modelling of that portion of the Browns Ferry fire in the Reactor Building indicates velocities nearly an order of magnitude larger than those reported; hence the neglect of fire growth inhibition by suppression efforts may add strong conservatism to the analysis. Finally, we comment on the problem of obtaining the unconditional frequency of fires which involve two cable trays. As defined above,J~ is the conditional frequency of fires involving two trays in the cable spreading room of a hypothetical plant, given a fire. To make this frequency unconditional, we must incorporate the frequency of fires in that particular room. While ref. 1 does provide a distribution for this quantity, some care must be exercised in using this information. The initial fire assumed in this analysis is moderately sized (1 ft2). Before we utilize any fire frequency distribution, we must decide if that distribution represents the frequency of fires which reach a comparable size (and thus are considered to be serious), or if it also includes the potentially large number of small fires which either

PROBABILISTIC MODELS FOR CABLE TRAY FIRES

227

self-extinguish or are put out immediately. We note that prior to the Browns Ferry fire, a number of small fires were initiated by the same leak test procedure which caused the large fire, but none of these were incorporated into the fire frequency data because they were small and were promptly extinguished.

ACKNOWLEDGEMENTS

This work was supported by the Probabilistic Analysis Staff of the United States Nuclear Regulatory Commission. The authors would like to thank Pickard, Lowe, and Garrick Inc., and the Commonwealth Edison Company for their support in the development of the cable spreading room application of the model presented. A special thanks also goes to Dr Mardyros Kazarians for his helpful comments and criticism.

REFERENCES 1. APOSTOLAKIS,G. and KAZARIANS,M. The frequency of fires in light water reactor compartments, ANS/ENS Topical Meeting on Thermal Reactor Safety, Knoxville, Tennessee, April 6-9, 1980. 2. SIu, N. O. Physical models for compartment fires, Reliability Engineering, 3(3) (1982), pp. 77-100. 3. KAPLAN, S. and GARRICK, n. J. On the quantitative definition of risk, Risk Analysis, 1(1981), pp. 11-27. 4. APOSTOLAKIS, G. Bayesian methods in.risk assessment. In: Advances in nuclear science and technology, J. Lewins and M. Becker (eds.), Plenum Press, New York, 1981. 5. PARRY,G. W. and WINTEa, P. W. Characterization and evaluation of uncertainty in probabilistic risk analysis, Nuclear Safety, 22 (Jan.-Feb. 1981), pp. 28-41. 6. APOSTOLAKIS,G. and MOSLEH,A. Expert opinion and statistical evidence: An application to reactor core melt frequency, Nuclear Sci. Eng., 70 (1979), pp. 135-49. 7. VAURIO, J. K. Response surface techniques developed for the probabilistic analysis of accident consequences, ANS/ENS Topical Meeting on Probabilistic Analysis of Nuclear Reactor Safety, Los Angeles, May 8-10, 1978. 8. IMAN, R. L., HELTON, J. C. and CAMPBELL, J. E. Risk methodology for geologic disposal of radioactive waste: Sensitivity analysis techniques, NUREG/CR-0394, SAND78-0912, October 1978. 9. Zion station fire protection report, Commonwealth Edison Company, April 30, 1980. 10. PINKEL,I. I. Estimating fire hazards within enclosed structures as related to nuclear power stations, BNL-23892, Brookhaven National Laboratory, January 1978. I 1. KAPLAN,S. On the method of discrete probability distributions in risk and reliability calculations, Risk Analysis (in press). 12. FLEMING,K. N., HOUGHTON,W. T. and SCALETTA,F. P. A methodology for risk assessment of major .]ires and its application to an HTGR plant, GA-A15402, General Atomic Company, 1979. 13. SCOTT,R. L. Browns Ferry nuclear power plant fire on Mar. 22, 1975, Nuclear Safety, 17 (Sept.-Oct. 1976), pp. 592-611.