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Numerical model of upward flame spread on practical wall materials
Fire Safety Journal 36 (2001) 225}240
Numerical model of upward #ame spread on practical wall materials E.G. Brehob *, C.I. Kim, A.K. Kulkarni Department of Mechanical Engineering, University of Louisville, Louisville, KY 40292, USA Chung-Ang University, South Korea The Pennsylvania State University, USA Received 30 September 1999; received in revised form 16 August 2000; accepted 11 October 2000
Abstract The focus of this paper is the development of a thermal, "nite di!erence numerical model to describe one-dimensional upward #ame spread on practical wall materials. Practical materials include composite materials and those that char, in addition to clean burning, homogeneous materials. A set of equations used in the model is developed and the methods for obtaining necessary `"re propertiesa are discussed. Some of the particular features of the model include the use of a correlation for #ame heat feedback and the use of an experimentally measured mass loss rate to incorporate the burning characteristics of practical materials. A comparison of the numerical predictions with the experimental results for #ame heights and temperatures are shown for Douglas "r particle board. The model correctly predicts trends but underpredicts the #ame heights and pyrolysis height in the cases tested. Two additional cases are shown for materials for which experimentally measured heat release rate data are used in place of the mass loss rate data. The #ame and pyrolysis height predictions are in much better agreement for these cases. Further e!orts to obtain material property data that is appropriate for #ame spread modeling is indicated by this work. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Upward #ame spread; Flame height; Mass loss rate
1. Introduction Upward #ame spread is the most rapid and hazardous of the various modes of #ame spread on a vertical wall, and consequently, of great interest in the "re safety
* Corresponding author. Tel.: #1-502-852-7852; fax: #1-502-852-6503. E-mail address: [email protected] (E.G. Brehob). 0379-7112/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 0 ) 0 0 0 5 4 - 0
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Nomenclature c C h h k K m n q q q q q QQ QQ
t T T T x x x x y
heat capacity (kJ/kg K) experimentally determined constant used in Eq. (3) heat of combustion (kJ/kg) heat transfer coe$cient (kW/m) thermal conductivity (kW/m K) experimentally determined constant used in Eq. (5) mass loss rate (kg/m s) experimentally determined constant used in Eq. (5) external radiation applied to sample surface (kW/m) heat feedback to sample surface due to #ames (kW/m) maximum heat feedback due to #ames (kW/m) radiative heat loss from sample surface (kW/m) sum of all heat transfer to sample surface (kW/m) energy release rate per unit length of the line burner (kW/m) energy release rate per unit length of material (kW/m) time (s) temperature (C) ignition temperature (C) initial sample temperature (C) distance from the leading edge of the sample (m) burnout front height (m) pyrolysis front height (m) #ame height (m) distance from surface to interior of sample (m) thermal di!usivity (m/s) density (kg/m)
"eld. Most compartment "res involve burning of vertical walls made of such combustible materials as wood panels, decorative plastic sheets, or composite materials, which exhibit combustion characteristics that are signi"cantly di!erent from an idealized semi-in"nite thick wall. The model presented here attempts to address the problem of upward #ame spread under external radiation to simulate #ame spread in an enclosure when surrounding #ames supply heat #ux to the wall of interest. It has been observed that certain materials which do not sustain upward #ame spread in the absence of external radiation #ux (for example, wood), allow the #ame to spread when assisted by an external radiation source [1]. Also, materials, which normally exhibit upward #ame spread will have a signi"cant enhancement in #ame spread and "re growth in the presence of external radiation [2,3]. These e!ects must be accounted for in the model. This paper presents results of a study on upward #ame spread over practical wall materials with and without external radiation heat #ux, with an
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emphasis on developing an analytical thermal model for one-dimensional upward #ame spread. The model is based on fundamental principles of #ame spread, but it also uses certain available experimental correlations and applied properties measured in independent small-scale experiments to account for important aspects of combustion of practical materials. To incorporate the combustion characteristics for materials that do not burn in an `analytically simplea way, a mass loss rate function for the material is experimentally measured and used in the model. A correlation to model heat feedback due to the #ames has been developed based on several practical materials. Experimental data on turbulent upward #ame spread is compared with numerical predictions to illustrate the capabilities of the model.
2. Previous work Attempts to model upward #ame spread have been made for many years, beginning with Thomas and others [4}6]. Most of these studies assumed `ideally behavinga materials such as those with constant forward heat #ux, uniform burning rate, and in"nite thickness. Hasemi [7] used a variable forward heat #ux to analyze temperature rise of the unburned fuel ahead of the pyrolysis front. Numerical and semiempirical algorithms have been discussed for the upward #ame spread problem with the emphasis on calculation of local pyrolysis rate [8,9]. Saito et al. [10] assumed a nonuniform local burning rate (m ) with "nite burnout time characteristic of thin walls. In their analytical model a Volterra-type integral equation was obtained after making certain assumptions, and based on that, they discussed the behavior of the wall "re under limiting conditions for two speci"c cases of m . Karlsson and Thomas [11] and Baroudi and Kokkala [12] did further work with the equation developed by Saito et al. They used several analytical models for the rate of heat release and determined regions of accelerating and decelerating #ame spread. Karlsson [13] eventually extended the solution of the Volterra-type integral equation as applied to #ame spread to allow any shape of mass loss rate curve to be used. It becomes clear from these papers the importance of determining material properties, and in particular, a representation of mass burning rate or rate of heat release. In the paper by Cleary and Quintiere [14], a framework for developing #ame spread predictions for use with practical materials is given with the objective of using relatively standardized material property data. The set of governing equations were developed in appropriate terms to require only material data which can be obtained from the LIFT and cone calorimeter tests. Flame spread models have been developed by Mitler and Steckler [15], Beyler et al. [16] and Kokkala et al. [17] that make use of property data from the cone calorimeter. All models have the goal of predicting upward #ame spread using a combination of: (1) basic equations that describe the physics of #ame spread, (2) empirical relations, and (3) experimentally measured property data. Each model does a reasonably good job of predicting various #ame spread parameters; rate of heat release, pyrolysis height, etc. A conclusion made by several of the research groups is that the models are sensitive to input property data and further investigation to determine appropriate input is necessary to produce more robust predictive
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capabilities. The model presented is similar to the models in [15}17] since all are thermal models, use a combination of analytical and empirical equations to model the process, and use numerical methods to solve the equations. But, the current model uses a di!erent empirical model for estimating the heat feedback from the #ame and provides for the mass loss rate at each location on the face of the burning surface to vary as a function of time.
3. Development of a mathematical model The physical system under consideration is depicted in Fig. 1. A slab of material is subjected to a known, external heat #ux on its face, and is ignited at the bottom using a line burner. Because of the heat #ux provided by #ames of the burner and the external heat #ux, the surface temperature of bottom section of the wall starts rising. When the surface temperature reaches a certain characteristic temperature, ¹ , the material starts pyrolyzing signi"cantly. Then, #ames from the burning of pyrolyzed fuel cover the solid fuel above the pyrolysis front (x ), which is heated by the energy
Fig. 1. Physical model.
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feedback, q (x, t), from the #ames and the external radiation sources. When unburned fuel heats up to ¹ , it starts pyrolyzing, the #ames grow taller, and thus, the process of upward #ame spread is continued. External radiation a!ects upward #ame spread primarily in two ways. The radiant heat #ux adds to the heat feedback from the #ame and causes the yet-unburned surface of the sample to heat up to the pyrolysis temperature quicker. Over the surface area that is already burning, the external radiation increases the mass loss rate of the sample, which in turn causes higher #ames. The important parameters and processes contributing to the upward #ame spread, therefore, are the heating of the unburned fuel above the pyrolysis edge, the total heat feedback to the unburned wall surface, q (x, t), the #ame tip height, x , and the local mass loss rate of the wall on the pyrolyzing surface, m . Thus, the model for describing the upward #ame spread process is developed in four major parts: (i) Heating of the yet-unpyrolyzed section of the wall: It is well established that the #ame spread process depends on how fast the unburned fuel ahead of the pyrolysis front can be heated to a critical temperature (or a range of temperatures) that causes signi"cant pyrolysis. The surface temperature as well as the inner temperature of the two-dimensional vertical wall, initially at the temperature of T , are obtained by solving the transient two-dimensional heat conduction equation with transient boundary conditions, ¹ ¹ 1 ¹ # " . x y t Initial condition: Boundary conditions:
!k
!k
¹ y ¹ y
¹ !k x
W W"
(1)
¹(x, y, 0)"¹ . ¹(x, 0, t)"¹
for x(x ,
"q (x, t) for x'x , "h(¹!¹ ), "!k
¹ x
"0, V V& where y"0 to D and x"0 to H represents the domain for calculations. The thermal conductivity k is an average value for the material. (In general, k can vary with the depth for laminated or composite materials and can change once a material has been charred. In the present work, a single value reported in literature is used for the materials studied.) With the time-dependent boundary conditions, use of one-dimensional or two-dimensional conduction requires about the same computation time. The two-dimensional equation is preferred because it gives slightly better accuracy (especially around the pyrolysis edge area where there is a temperature gradient in the x direction), and easier de"nition of computation domain. It is assumed that the
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ignition temperature does not vary with location, the material is inert during the heating process, and the heat loss to the backside of the wall is in the form of convection to the ambient atmosphere. (ii) Total heat feedback, qR (x, t): This is the heat feedback to the wall above the pyrolysis front and it is responsible for raising the surface temperature to some critical value which causes signi"cant pyrolysis. It is modeled as q "q #q !q ,
(2)
where q is the #ame heat feedback, q is the external incident radiation, and q is the heat loss because of surface radiation. The external radiation is known and the reradiation may be calculated since the sample surface temperature and ambient surrounding temperature are known. In some of the previous studies, q (x, t) has been assumed to be constant, typically equal to 25 or 30 kW/m, up to a certain height and then zero above [9]. Analytical derivation of q (x, t) involves use of several thermal/physical properties in order to estimate the heat #ux for a given material. However, the properties are often not available, or the available properties are not known with su$cient accuracy. Therefore, in the present work, q (x, t) was obtained from measured data in order to allow its adequate representation. Fig. 2 shows the heat feedback data for black PMMA, clear PMMA, cardboard, Douglas "r particle board, rigid polyurethane foam, and carpet [18,19]. The best "t function for these sets of data when expressed in the
Fig. 2. Correlations for heat feedback using data from [18,19].
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Fig. 3. Flame heat feedback data for particle board.
appropriate dimensionless variable was an exponential decay, given by
x!x , q (x, t)"q exp !C (3) x !x where q is the maximum heat feedback from #ames to surface and the decay factor C was determined to be !1.37. The above correlation allows determination of q based on a single value of q which may be treated as a `"re propertya of the wall material. An example of the data used for determining the maximum forward heat feedback can be seen in Fig. 3. The "gure shows the transient heat feedback to the surface of a burning sample, in this case Douglas "r particle board, with an external radiation #ux of 2.2 kW/m. At each height on the sample face, the heat #ux reaches a maximum plateau value. Calculating the average plateau value for all gages and subtracting the external radiation yields the maximum forward heat feedback due to the #ames. The value of q for Douglas "r particle board is 32 kW/m as seen in Fig. 3. Maximum forward heat feedback was measured for other charring materials (such as plywood, cardboard, poplar) and the values are within 30% of this value. A constant value has been chosen, but it has been suggested [20] that the maximum heat feedback may increase as the #ames grow. Over the heights tested in this work, a 1.2 m sample, the maximum forward heat feedback did not appear to increase signi"cantly. (iii) Instantaneous yame height: The equation for forward heat #ux requires an expression for #ame height, x . Flame size depends on the cumulative energy release rate, which may be calculated by the integration of local burning rate over the entire
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pyrolyzing region on the wall. The larger the total energy release rate, the taller the #ames are, and a greater area ahead of the pyrolysis front is preheated. The #ame tip height is obtained from an available correlation [7], x (t)!x (t)"K[QQ Y #QQ Y ]L, (4)
where the bracket represents the total energy release rate per unit width in the "re which is the sum of the energy release rate per width by the igniter (QQ Y in kW/m) and the energy release rate per width due to the pyrolyzing surface (QQ Y in kW/m).
Experimental correlation factors K and n were determined as 0.0433 and 2/3, respectively [21] to result in #ame heights given in meters. (iv) Transient local mass loss or burning rate, m (t): To close the model, an estimate of total energy release rate is needed. The total energy release rate at any instant is obtained from the total mass loss rate of the burning slab multiplied by the average heat of combustion of pyrolyzed fuel,
V R m dx. (5) V R The total or cumulative mass loss rate is calculated by integrating the local mass loss rate over the entire burning area. From the #ame spread point of view, the history of local burning rate is a characteristic of the combustion behavior of the material and it depends on the physical, chemical and geometric properties of the wall material. For a slab of a homogeneous material like PMMA, the rate of burning increases in the beginning because the heat loss to the interior decreases initially, goes through a maximum burning rate, and "nally drops to zero due to heat loss from the back side of "nite thickness samples [22]. In the simplest form, m (t) has been assumed to be constant until all the combustible mass is pyrolyzed; other forms have employed inverse square root or exponentially decreasing dependence. During the #ame spread process diwerent vertical locations are at diwerent stages of burning, for example, the region near the trailing edge may be close to burnout, the region farther up may be burning at the peak rate, and the pyrolysis edge is at the just-ignited state. The history of burning is therefore very important in the #ame spread process. The external radiation adds to the #ame heat feedback and enhances the burning rate. In the #ame spread model by Mitler and Steckler [15], an experimentally measured mass loss rate as measured by the cone calorimeter was used. They incorporated an accelerating parameter to account for the experimental mass loss rate data being available at only a speci"c #ame heat #ux and the model requires this information at #ame heat #uxes other this. A calculation of the heat of gasi"cation (not assumed a constant value) is also made in their model. In the present work m (t) is obtained from separate small-scale mass loss rate experiments. During the upward #ame spreading process, the unburned fuel just above the pyrolysis front is continuously covered by turbulent #ames emerging from the pyrolyzing region below. Therefore, the local, transient mass loss rate measured while a material sample is continuously covered by turbulent #ames and subjected to the external radiation heat, is taken to be the representative combustion behavior of QQ ' "h
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Fig. 4. Mass loss rate curves for Douglas "r particle board at three levels of external radiation with data taken from Parker [23].
the material and it is expected to be only weakly dependent on the location. Quantifying the energy release rate experimentally appeared most reasonable for complex materials that burn with their own `signaturea. Fig. 4 shows the mass loss rate curves derived from experiments at three levels of external radiation for a typical wood-based material, in this case particle board. These local mass loss rate functions are used in the present calculations. Data from Parker [23] is shown in this "gure. The data was originally presented as heat release rate and has been converted to mass loss rate by dividing by a constant heat of combustion. Although the data is for particle board at much higher external radiation, the main feature of the two-peaked mass loss rate is evident. For further details on these experiments and their justi"cation for use in #ame spread models, refer to [24]. Eqs. (1)}(5) form a complete set of equations with appropriate boundary conditions. The general solution procedure is as follows: at any given instant, t, "rst the local burning rate, m , is computed; it is then integrated over x"x to x ; then the total heat release rate, #ame height, and q (x, t) are computed; the temperature distribution in the solid is computed to obtain the surface temperature distribution; "nally, the new x is determined based on the condition T"T at the surface. The computation is started at the instant the burner is placed near the bottom of the wall. Equations are solved using a "nite di!erence, Gauss}Seidel iterative numerical procedure. Because of sti! temperature gradients close to the surface, a nonuniform grid is used with a higher grid density near the surface.
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4. Experiments Upward #ame spread experiments were conducted on a small but turbulent scale apparatus which allowed testing of 30 cm wide;120 cm high samples. Measurements on turbulent #ame height and pyrolysis height as a function of time were made for various practical materials. Samples were #ush mounted in a larger backboard of inert material to ensure a two-dimensional #ame as much as possible and to maintain a continuous #ame well over the sample height. Distance from the leading edge of the sample was marked on the surrounding walls to measure #ame height and pyrolysis height. A high shutter-speed video camera was used to record the #ame propagation process to measure #ame height and pyrolysis height at various stages of the spread. Gardon foil heat #ux gages were used to obtain information on the heat feedback to the surface and thermocouples were installed on the front and back fuel surfaces for temperature data. Additional details on the experimental setup, instrumentation, and test procedure are given in [25].
5. Discussion and results Model calculations and experimental data for #ame spread are presented in Fig. 5 for 15.9 mm Douglas "r particle board. Particle board is a relatively homogeneous wood-based material with a binder. Flame height was based on the top most visible tip of the #ame from a single frame of video. Sporadic, detached #amelets less than 5 cm in size were ignored. Ten consecutive frames were read at the beginning of each time step and the average and #uctuation were determined. The same technique was applied to all #ame height measurements. The pyrolysis height was the most di$cult to measure. A researcher with a side view of the burning sample held a pointer to locate the pyrolysis height. Although #ames may extend above the height of the sample, #ame spread was not classi"ed as sustained unless the pyrolysis front also spread to the top of the sample. Property data required for the numerical model are shown in Table 1. Some of the properties were measured in the laboratory (, m , and q ) and others (h , c, k, and T ) were obtained from literature [10,17,26,27]. The range of values found for the combination of properties called thermal inertia (k c) spanned an order of magnitude. k and c for wood-based materials are known to vary with moisture content and temperature, and may account for some of the variation of thermal inertia values found in literature. Flame spread heights as a function of time are shown in Fig. 5 for 15.9 mm thick Douglas "r particle board. In this set of experimental tests, #ames spread to the top of the sample for the two levels of external radiation, but were not sustained to the top when no external radiation was supplied. The numerical predictions of #ame height show the #ame spread to be sustained to the top of the sample for all cases. In general, the numerical calculations underpredict the #ame heights. To investigate the sensitivity of the upward #ame spread model to property data; several properties of Douglas "r particle board were varied. h and q were increased by 20% of the values used in
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Fig. 5. Comparison of model predictions with experimental data for #ame height on Douglas "r particle board at di!erent levels of external radiation.
Table 1 Material properties used in numerical models Material properties
Units
Particle board (US)
Particle board (Japan)
q ¹ k c kc h Q Thickness
kW/m C kW/m K kg/m kJ/kgK (kW/m K)s kJ/kg kW/m mm
31.7 320 2.02E}04 767 4.04 0.6259 15,000 18.5 15.9
31.7 320 2.50E}04 780 4.8 0.936 n/a 18.5 12.6
the base case, while k and T was decreased by 20%. All four properties were varied such that an increase in #ame spread would be expected. Although the numerically determined #ame height was in better agreement with the experimental data, the #ame height was still underpredicted over most of the sample. Many de"nitions for selection of #ame height can be used and often comparisons of pyrolysis front may be used to obtain a more reliable assessment of the performance of a numerical prediction. By using the conventional assumption that the pyrolysis front is the highest location on the fuel surface that has reached ignition temperature, knowledge of surface temperature can readily be used to determine the pyrolysis front. In Fig. 6, experimentally
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Fig. 6. Temperatures at several heights on the surface of Douglas "r particle board with 7.6 kW/m of external radiation applied.
Fig. 7. Comparison of model predictions with experimental data for a composite material * "gure taken from Ohlemiller and Cleary [28].
measured and numerically predicted surface temperatures are plotted for 15.9 mm particle board. At a height of 28 cm from the lower edge of the sample face, the model does a good job of predicting the surface temperature as a function of time. The numerical model signi"cantly underpredicts the temperature at all higher locations on the sample.
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The model does exhibit the correct trends (#ame heights increase as external radiation is increased), but the predicted #ame heights do not correspond with those measured experimentally. The authors consider obtaining appropriate property data to be the major di$culty in using the model. Two examples will be used to illustrate this proposal. The numerical model was run independently by Ohlemiller and Cleary [28] for a composite material made of epoxy and "berglass; and the model predictions were compared with their experimental #ame spread data (Fig. 7). The material properties were measured for the speci"c material, not obtained from literature. Experimentally measured heat release rate data was used, as opposed to the mass loss rate data and assumption of a constant heat of combustion. The results for the composite material showed good agreement between experimental measurement and prediction of pyrolysis height. This suggests that property data obtained for exactly the same type of material may be needed for accurate prediction of #ame spread rates. The #ame spread model was also run using experimentally measured heat release rate data. The experimental data for #ame and pyrolysis height was taken from Hasemi et al. [29] and is shown in Fig. 8 and is based on 12.6 mm thick particle board at 6.0 kW/m of external radiation. Heat release rate data "t with an exponential decay function was also taken from Hasemi et al. [29] for the same material. An equation of the form
Q "
V R q exp(!t) dx, V R
(6)
was used to model the experimental heat release rate information. q of 180 kW/m and "0.005/s was used. Time, t, is a local time for each element of fuel that has begun to pyrolyze. The material properties used in the numerical model are given in Table 1. Also note that an alternate value of the constant K, found in Eq. (4), has been used for this case, as suggested by Hasemi et al. (K"0.056). Model predictions for #ame height and pyrolysis height are in good agreement with the experimental data. For charring materials, such as wood, using experimental mass loss rate data and assuming a constant heat of combustion throughout the burning process may not be a realistic model of heat release rate. Volatiles which supply more heating energy per unit mass of fuel may be pyrolyzed earlier on, while the remaining char may continue to be consumed but with much less of a contribution of fuel energy to the #ames.
6. Conclusion In the current model, emphasis has been placed on developing a practical model that will be useful for a broad range of materials and may be part of a larger room "re model. Some of the important aspects of the model are: (1) the use of a correlation for #ame Additional data (actual numerical values of certain parameters) were supplied by Hasemi and is not included in this reference.
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Fig. 8. Comparison of model predictions with experimental data for the particle board * data taken from Hasemi et al. [29].
heat feedback distribution to the sample surface based on data available in the literature, and (2) the use of an experimentally measured mass loss rate or rate of heat release for the sample material. The development of the mathematical model is presented and #ame spread predictions obtained using the model are compared with experimental results. The model appears to capture the physics of the upward #ame spread process, for example, the #ame tip predictions as a function of time demonstrate correct trends and often correctly predicted whether sustained #ame spread occurred. Using an experimentally measured mass loss rate in the numerical model resulted in estimates of the #ame spread rates that were lower than those experimentally measured. Several other wood-based materials, in addition to the particle board data presented, exhibited this same trend. The use of experimentally measured heat release rate data signi"cantly improved the agreement between predicted and measured #ame and pyrolysis heights but tended to overpredict the #ame spread rates. Based on this work it is suggested that heat release rate data be used in the numerical model as it results in higher #ame spread rate predictions, which is desirable in "re safety calculations. In order to obtain more reliable predictions of #ame and pyrolysis heights, a need exists for thermal property data for practical materials that will be appropriate for #ame spread models.
Acknowledgements This research was supported by the Building and Fire Research Laboratory of the National Institute of Standards and Technology under grant no. NANB8D0849.
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[24] Brehob EG, Kulkarni AK. Time-dependent mass loss rate behavior of wall materials under external radiation. Fire Mater 1993; 17 (5): 249}54. [25] Brehob EG, Kulkarni AK. Experimental measurements of upward #ame spread on a vertical wall with external radiation. Fire Safety J 1998;31:181}200. [26] Fangrat J. Di!erent approach to some "re modeling input data. Heat Release and Fire Hazard * Proceedings of the First Japan Symposium, Tsukuba, Japan, vol. 1, II-1 to II-6, 1993. [27] Incropera FP, DeWitt DP. Fundamentals of heat transfer. New York: Wiley, 1985. p. 763}90. [28] Ohlemiller TJ, Cleary TG. Upward #ame spread on composite materials. Fire Safety J 1999;32:159}72. [29] Hasemi Y, Yoshida M, Yasui N, Parker WJ. Upward #ame spread along a vertical solid for a transient local heat release rate. Fire Safety Science * Proceedings of the Fourth International Symposium, 1994. p. 385}96.
Numerical model of upward #ame spread on practical wall materials E.G. Brehob *, C.I. Kim, A.K. Kulkarni Department of Mechanical Engineering, University of Louisville, Louisville, KY 40292, USA Chung-Ang University, South Korea The Pennsylvania State University, USA Received 30 September 1999; received in revised form 16 August 2000; accepted 11 October 2000
Abstract The focus of this paper is the development of a thermal, "nite di!erence numerical model to describe one-dimensional upward #ame spread on practical wall materials. Practical materials include composite materials and those that char, in addition to clean burning, homogeneous materials. A set of equations used in the model is developed and the methods for obtaining necessary `"re propertiesa are discussed. Some of the particular features of the model include the use of a correlation for #ame heat feedback and the use of an experimentally measured mass loss rate to incorporate the burning characteristics of practical materials. A comparison of the numerical predictions with the experimental results for #ame heights and temperatures are shown for Douglas "r particle board. The model correctly predicts trends but underpredicts the #ame heights and pyrolysis height in the cases tested. Two additional cases are shown for materials for which experimentally measured heat release rate data are used in place of the mass loss rate data. The #ame and pyrolysis height predictions are in much better agreement for these cases. Further e!orts to obtain material property data that is appropriate for #ame spread modeling is indicated by this work. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Upward #ame spread; Flame height; Mass loss rate
1. Introduction Upward #ame spread is the most rapid and hazardous of the various modes of #ame spread on a vertical wall, and consequently, of great interest in the "re safety
* Corresponding author. Tel.: #1-502-852-7852; fax: #1-502-852-6503. E-mail address: [email protected] (E.G. Brehob). 0379-7112/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 0 ) 0 0 0 5 4 - 0
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Nomenclature c C h h k K m n q q q q q QQ QQ
t T T T x x x x y
heat capacity (kJ/kg K) experimentally determined constant used in Eq. (3) heat of combustion (kJ/kg) heat transfer coe$cient (kW/m) thermal conductivity (kW/m K) experimentally determined constant used in Eq. (5) mass loss rate (kg/m s) experimentally determined constant used in Eq. (5) external radiation applied to sample surface (kW/m) heat feedback to sample surface due to #ames (kW/m) maximum heat feedback due to #ames (kW/m) radiative heat loss from sample surface (kW/m) sum of all heat transfer to sample surface (kW/m) energy release rate per unit length of the line burner (kW/m) energy release rate per unit length of material (kW/m) time (s) temperature (C) ignition temperature (C) initial sample temperature (C) distance from the leading edge of the sample (m) burnout front height (m) pyrolysis front height (m) #ame height (m) distance from surface to interior of sample (m) thermal di!usivity (m/s) density (kg/m)
"eld. Most compartment "res involve burning of vertical walls made of such combustible materials as wood panels, decorative plastic sheets, or composite materials, which exhibit combustion characteristics that are signi"cantly di!erent from an idealized semi-in"nite thick wall. The model presented here attempts to address the problem of upward #ame spread under external radiation to simulate #ame spread in an enclosure when surrounding #ames supply heat #ux to the wall of interest. It has been observed that certain materials which do not sustain upward #ame spread in the absence of external radiation #ux (for example, wood), allow the #ame to spread when assisted by an external radiation source [1]. Also, materials, which normally exhibit upward #ame spread will have a signi"cant enhancement in #ame spread and "re growth in the presence of external radiation [2,3]. These e!ects must be accounted for in the model. This paper presents results of a study on upward #ame spread over practical wall materials with and without external radiation heat #ux, with an
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emphasis on developing an analytical thermal model for one-dimensional upward #ame spread. The model is based on fundamental principles of #ame spread, but it also uses certain available experimental correlations and applied properties measured in independent small-scale experiments to account for important aspects of combustion of practical materials. To incorporate the combustion characteristics for materials that do not burn in an `analytically simplea way, a mass loss rate function for the material is experimentally measured and used in the model. A correlation to model heat feedback due to the #ames has been developed based on several practical materials. Experimental data on turbulent upward #ame spread is compared with numerical predictions to illustrate the capabilities of the model.
2. Previous work Attempts to model upward #ame spread have been made for many years, beginning with Thomas and others [4}6]. Most of these studies assumed `ideally behavinga materials such as those with constant forward heat #ux, uniform burning rate, and in"nite thickness. Hasemi [7] used a variable forward heat #ux to analyze temperature rise of the unburned fuel ahead of the pyrolysis front. Numerical and semiempirical algorithms have been discussed for the upward #ame spread problem with the emphasis on calculation of local pyrolysis rate [8,9]. Saito et al. [10] assumed a nonuniform local burning rate (m ) with "nite burnout time characteristic of thin walls. In their analytical model a Volterra-type integral equation was obtained after making certain assumptions, and based on that, they discussed the behavior of the wall "re under limiting conditions for two speci"c cases of m . Karlsson and Thomas [11] and Baroudi and Kokkala [12] did further work with the equation developed by Saito et al. They used several analytical models for the rate of heat release and determined regions of accelerating and decelerating #ame spread. Karlsson [13] eventually extended the solution of the Volterra-type integral equation as applied to #ame spread to allow any shape of mass loss rate curve to be used. It becomes clear from these papers the importance of determining material properties, and in particular, a representation of mass burning rate or rate of heat release. In the paper by Cleary and Quintiere [14], a framework for developing #ame spread predictions for use with practical materials is given with the objective of using relatively standardized material property data. The set of governing equations were developed in appropriate terms to require only material data which can be obtained from the LIFT and cone calorimeter tests. Flame spread models have been developed by Mitler and Steckler [15], Beyler et al. [16] and Kokkala et al. [17] that make use of property data from the cone calorimeter. All models have the goal of predicting upward #ame spread using a combination of: (1) basic equations that describe the physics of #ame spread, (2) empirical relations, and (3) experimentally measured property data. Each model does a reasonably good job of predicting various #ame spread parameters; rate of heat release, pyrolysis height, etc. A conclusion made by several of the research groups is that the models are sensitive to input property data and further investigation to determine appropriate input is necessary to produce more robust predictive
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capabilities. The model presented is similar to the models in [15}17] since all are thermal models, use a combination of analytical and empirical equations to model the process, and use numerical methods to solve the equations. But, the current model uses a di!erent empirical model for estimating the heat feedback from the #ame and provides for the mass loss rate at each location on the face of the burning surface to vary as a function of time.
3. Development of a mathematical model The physical system under consideration is depicted in Fig. 1. A slab of material is subjected to a known, external heat #ux on its face, and is ignited at the bottom using a line burner. Because of the heat #ux provided by #ames of the burner and the external heat #ux, the surface temperature of bottom section of the wall starts rising. When the surface temperature reaches a certain characteristic temperature, ¹ , the material starts pyrolyzing signi"cantly. Then, #ames from the burning of pyrolyzed fuel cover the solid fuel above the pyrolysis front (x ), which is heated by the energy
Fig. 1. Physical model.
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feedback, q (x, t), from the #ames and the external radiation sources. When unburned fuel heats up to ¹ , it starts pyrolyzing, the #ames grow taller, and thus, the process of upward #ame spread is continued. External radiation a!ects upward #ame spread primarily in two ways. The radiant heat #ux adds to the heat feedback from the #ame and causes the yet-unburned surface of the sample to heat up to the pyrolysis temperature quicker. Over the surface area that is already burning, the external radiation increases the mass loss rate of the sample, which in turn causes higher #ames. The important parameters and processes contributing to the upward #ame spread, therefore, are the heating of the unburned fuel above the pyrolysis edge, the total heat feedback to the unburned wall surface, q (x, t), the #ame tip height, x , and the local mass loss rate of the wall on the pyrolyzing surface, m . Thus, the model for describing the upward #ame spread process is developed in four major parts: (i) Heating of the yet-unpyrolyzed section of the wall: It is well established that the #ame spread process depends on how fast the unburned fuel ahead of the pyrolysis front can be heated to a critical temperature (or a range of temperatures) that causes signi"cant pyrolysis. The surface temperature as well as the inner temperature of the two-dimensional vertical wall, initially at the temperature of T , are obtained by solving the transient two-dimensional heat conduction equation with transient boundary conditions, ¹ ¹ 1 ¹ # " . x y t Initial condition: Boundary conditions:
!k
!k
¹ y ¹ y
¹ !k x
W W"
(1)
¹(x, y, 0)"¹ . ¹(x, 0, t)"¹
for x(x ,
"q (x, t) for x'x , "h(¹!¹ ), "!k
¹ x
"0, V V& where y"0 to D and x"0 to H represents the domain for calculations. The thermal conductivity k is an average value for the material. (In general, k can vary with the depth for laminated or composite materials and can change once a material has been charred. In the present work, a single value reported in literature is used for the materials studied.) With the time-dependent boundary conditions, use of one-dimensional or two-dimensional conduction requires about the same computation time. The two-dimensional equation is preferred because it gives slightly better accuracy (especially around the pyrolysis edge area where there is a temperature gradient in the x direction), and easier de"nition of computation domain. It is assumed that the
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ignition temperature does not vary with location, the material is inert during the heating process, and the heat loss to the backside of the wall is in the form of convection to the ambient atmosphere. (ii) Total heat feedback, qR (x, t): This is the heat feedback to the wall above the pyrolysis front and it is responsible for raising the surface temperature to some critical value which causes signi"cant pyrolysis. It is modeled as q "q #q !q ,
(2)
where q is the #ame heat feedback, q is the external incident radiation, and q is the heat loss because of surface radiation. The external radiation is known and the reradiation may be calculated since the sample surface temperature and ambient surrounding temperature are known. In some of the previous studies, q (x, t) has been assumed to be constant, typically equal to 25 or 30 kW/m, up to a certain height and then zero above [9]. Analytical derivation of q (x, t) involves use of several thermal/physical properties in order to estimate the heat #ux for a given material. However, the properties are often not available, or the available properties are not known with su$cient accuracy. Therefore, in the present work, q (x, t) was obtained from measured data in order to allow its adequate representation. Fig. 2 shows the heat feedback data for black PMMA, clear PMMA, cardboard, Douglas "r particle board, rigid polyurethane foam, and carpet [18,19]. The best "t function for these sets of data when expressed in the
Fig. 2. Correlations for heat feedback using data from [18,19].
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Fig. 3. Flame heat feedback data for particle board.
appropriate dimensionless variable was an exponential decay, given by
x!x , q (x, t)"q exp !C (3) x !x where q is the maximum heat feedback from #ames to surface and the decay factor C was determined to be !1.37. The above correlation allows determination of q based on a single value of q which may be treated as a `"re propertya of the wall material. An example of the data used for determining the maximum forward heat feedback can be seen in Fig. 3. The "gure shows the transient heat feedback to the surface of a burning sample, in this case Douglas "r particle board, with an external radiation #ux of 2.2 kW/m. At each height on the sample face, the heat #ux reaches a maximum plateau value. Calculating the average plateau value for all gages and subtracting the external radiation yields the maximum forward heat feedback due to the #ames. The value of q for Douglas "r particle board is 32 kW/m as seen in Fig. 3. Maximum forward heat feedback was measured for other charring materials (such as plywood, cardboard, poplar) and the values are within 30% of this value. A constant value has been chosen, but it has been suggested [20] that the maximum heat feedback may increase as the #ames grow. Over the heights tested in this work, a 1.2 m sample, the maximum forward heat feedback did not appear to increase signi"cantly. (iii) Instantaneous yame height: The equation for forward heat #ux requires an expression for #ame height, x . Flame size depends on the cumulative energy release rate, which may be calculated by the integration of local burning rate over the entire
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pyrolyzing region on the wall. The larger the total energy release rate, the taller the #ames are, and a greater area ahead of the pyrolysis front is preheated. The #ame tip height is obtained from an available correlation [7], x (t)!x (t)"K[QQ Y #QQ Y ]L, (4)
where the bracket represents the total energy release rate per unit width in the "re which is the sum of the energy release rate per width by the igniter (QQ Y in kW/m) and the energy release rate per width due to the pyrolyzing surface (QQ Y in kW/m).
Experimental correlation factors K and n were determined as 0.0433 and 2/3, respectively [21] to result in #ame heights given in meters. (iv) Transient local mass loss or burning rate, m (t): To close the model, an estimate of total energy release rate is needed. The total energy release rate at any instant is obtained from the total mass loss rate of the burning slab multiplied by the average heat of combustion of pyrolyzed fuel,
V R m dx. (5) V R The total or cumulative mass loss rate is calculated by integrating the local mass loss rate over the entire burning area. From the #ame spread point of view, the history of local burning rate is a characteristic of the combustion behavior of the material and it depends on the physical, chemical and geometric properties of the wall material. For a slab of a homogeneous material like PMMA, the rate of burning increases in the beginning because the heat loss to the interior decreases initially, goes through a maximum burning rate, and "nally drops to zero due to heat loss from the back side of "nite thickness samples [22]. In the simplest form, m (t) has been assumed to be constant until all the combustible mass is pyrolyzed; other forms have employed inverse square root or exponentially decreasing dependence. During the #ame spread process diwerent vertical locations are at diwerent stages of burning, for example, the region near the trailing edge may be close to burnout, the region farther up may be burning at the peak rate, and the pyrolysis edge is at the just-ignited state. The history of burning is therefore very important in the #ame spread process. The external radiation adds to the #ame heat feedback and enhances the burning rate. In the #ame spread model by Mitler and Steckler [15], an experimentally measured mass loss rate as measured by the cone calorimeter was used. They incorporated an accelerating parameter to account for the experimental mass loss rate data being available at only a speci"c #ame heat #ux and the model requires this information at #ame heat #uxes other this. A calculation of the heat of gasi"cation (not assumed a constant value) is also made in their model. In the present work m (t) is obtained from separate small-scale mass loss rate experiments. During the upward #ame spreading process, the unburned fuel just above the pyrolysis front is continuously covered by turbulent #ames emerging from the pyrolyzing region below. Therefore, the local, transient mass loss rate measured while a material sample is continuously covered by turbulent #ames and subjected to the external radiation heat, is taken to be the representative combustion behavior of QQ ' "h
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Fig. 4. Mass loss rate curves for Douglas "r particle board at three levels of external radiation with data taken from Parker [23].
the material and it is expected to be only weakly dependent on the location. Quantifying the energy release rate experimentally appeared most reasonable for complex materials that burn with their own `signaturea. Fig. 4 shows the mass loss rate curves derived from experiments at three levels of external radiation for a typical wood-based material, in this case particle board. These local mass loss rate functions are used in the present calculations. Data from Parker [23] is shown in this "gure. The data was originally presented as heat release rate and has been converted to mass loss rate by dividing by a constant heat of combustion. Although the data is for particle board at much higher external radiation, the main feature of the two-peaked mass loss rate is evident. For further details on these experiments and their justi"cation for use in #ame spread models, refer to [24]. Eqs. (1)}(5) form a complete set of equations with appropriate boundary conditions. The general solution procedure is as follows: at any given instant, t, "rst the local burning rate, m , is computed; it is then integrated over x"x to x ; then the total heat release rate, #ame height, and q (x, t) are computed; the temperature distribution in the solid is computed to obtain the surface temperature distribution; "nally, the new x is determined based on the condition T"T at the surface. The computation is started at the instant the burner is placed near the bottom of the wall. Equations are solved using a "nite di!erence, Gauss}Seidel iterative numerical procedure. Because of sti! temperature gradients close to the surface, a nonuniform grid is used with a higher grid density near the surface.
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4. Experiments Upward #ame spread experiments were conducted on a small but turbulent scale apparatus which allowed testing of 30 cm wide;120 cm high samples. Measurements on turbulent #ame height and pyrolysis height as a function of time were made for various practical materials. Samples were #ush mounted in a larger backboard of inert material to ensure a two-dimensional #ame as much as possible and to maintain a continuous #ame well over the sample height. Distance from the leading edge of the sample was marked on the surrounding walls to measure #ame height and pyrolysis height. A high shutter-speed video camera was used to record the #ame propagation process to measure #ame height and pyrolysis height at various stages of the spread. Gardon foil heat #ux gages were used to obtain information on the heat feedback to the surface and thermocouples were installed on the front and back fuel surfaces for temperature data. Additional details on the experimental setup, instrumentation, and test procedure are given in [25].
5. Discussion and results Model calculations and experimental data for #ame spread are presented in Fig. 5 for 15.9 mm Douglas "r particle board. Particle board is a relatively homogeneous wood-based material with a binder. Flame height was based on the top most visible tip of the #ame from a single frame of video. Sporadic, detached #amelets less than 5 cm in size were ignored. Ten consecutive frames were read at the beginning of each time step and the average and #uctuation were determined. The same technique was applied to all #ame height measurements. The pyrolysis height was the most di$cult to measure. A researcher with a side view of the burning sample held a pointer to locate the pyrolysis height. Although #ames may extend above the height of the sample, #ame spread was not classi"ed as sustained unless the pyrolysis front also spread to the top of the sample. Property data required for the numerical model are shown in Table 1. Some of the properties were measured in the laboratory (, m , and q ) and others (h , c, k, and T ) were obtained from literature [10,17,26,27]. The range of values found for the combination of properties called thermal inertia (k c) spanned an order of magnitude. k and c for wood-based materials are known to vary with moisture content and temperature, and may account for some of the variation of thermal inertia values found in literature. Flame spread heights as a function of time are shown in Fig. 5 for 15.9 mm thick Douglas "r particle board. In this set of experimental tests, #ames spread to the top of the sample for the two levels of external radiation, but were not sustained to the top when no external radiation was supplied. The numerical predictions of #ame height show the #ame spread to be sustained to the top of the sample for all cases. In general, the numerical calculations underpredict the #ame heights. To investigate the sensitivity of the upward #ame spread model to property data; several properties of Douglas "r particle board were varied. h and q were increased by 20% of the values used in
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Fig. 5. Comparison of model predictions with experimental data for #ame height on Douglas "r particle board at di!erent levels of external radiation.
Table 1 Material properties used in numerical models Material properties
Units
Particle board (US)
Particle board (Japan)
q ¹ k c kc h Q Thickness
kW/m C kW/m K kg/m kJ/kgK (kW/m K)s kJ/kg kW/m mm
31.7 320 2.02E}04 767 4.04 0.6259 15,000 18.5 15.9
31.7 320 2.50E}04 780 4.8 0.936 n/a 18.5 12.6
the base case, while k and T was decreased by 20%. All four properties were varied such that an increase in #ame spread would be expected. Although the numerically determined #ame height was in better agreement with the experimental data, the #ame height was still underpredicted over most of the sample. Many de"nitions for selection of #ame height can be used and often comparisons of pyrolysis front may be used to obtain a more reliable assessment of the performance of a numerical prediction. By using the conventional assumption that the pyrolysis front is the highest location on the fuel surface that has reached ignition temperature, knowledge of surface temperature can readily be used to determine the pyrolysis front. In Fig. 6, experimentally
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Fig. 6. Temperatures at several heights on the surface of Douglas "r particle board with 7.6 kW/m of external radiation applied.
Fig. 7. Comparison of model predictions with experimental data for a composite material * "gure taken from Ohlemiller and Cleary [28].
measured and numerically predicted surface temperatures are plotted for 15.9 mm particle board. At a height of 28 cm from the lower edge of the sample face, the model does a good job of predicting the surface temperature as a function of time. The numerical model signi"cantly underpredicts the temperature at all higher locations on the sample.
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The model does exhibit the correct trends (#ame heights increase as external radiation is increased), but the predicted #ame heights do not correspond with those measured experimentally. The authors consider obtaining appropriate property data to be the major di$culty in using the model. Two examples will be used to illustrate this proposal. The numerical model was run independently by Ohlemiller and Cleary [28] for a composite material made of epoxy and "berglass; and the model predictions were compared with their experimental #ame spread data (Fig. 7). The material properties were measured for the speci"c material, not obtained from literature. Experimentally measured heat release rate data was used, as opposed to the mass loss rate data and assumption of a constant heat of combustion. The results for the composite material showed good agreement between experimental measurement and prediction of pyrolysis height. This suggests that property data obtained for exactly the same type of material may be needed for accurate prediction of #ame spread rates. The #ame spread model was also run using experimentally measured heat release rate data. The experimental data for #ame and pyrolysis height was taken from Hasemi et al. [29] and is shown in Fig. 8 and is based on 12.6 mm thick particle board at 6.0 kW/m of external radiation. Heat release rate data "t with an exponential decay function was also taken from Hasemi et al. [29] for the same material. An equation of the form
Q "
V R q exp(!t) dx, V R
(6)
was used to model the experimental heat release rate information. q of 180 kW/m and "0.005/s was used. Time, t, is a local time for each element of fuel that has begun to pyrolyze. The material properties used in the numerical model are given in Table 1. Also note that an alternate value of the constant K, found in Eq. (4), has been used for this case, as suggested by Hasemi et al. (K"0.056). Model predictions for #ame height and pyrolysis height are in good agreement with the experimental data. For charring materials, such as wood, using experimental mass loss rate data and assuming a constant heat of combustion throughout the burning process may not be a realistic model of heat release rate. Volatiles which supply more heating energy per unit mass of fuel may be pyrolyzed earlier on, while the remaining char may continue to be consumed but with much less of a contribution of fuel energy to the #ames.
6. Conclusion In the current model, emphasis has been placed on developing a practical model that will be useful for a broad range of materials and may be part of a larger room "re model. Some of the important aspects of the model are: (1) the use of a correlation for #ame Additional data (actual numerical values of certain parameters) were supplied by Hasemi and is not included in this reference.
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Fig. 8. Comparison of model predictions with experimental data for the particle board * data taken from Hasemi et al. [29].
heat feedback distribution to the sample surface based on data available in the literature, and (2) the use of an experimentally measured mass loss rate or rate of heat release for the sample material. The development of the mathematical model is presented and #ame spread predictions obtained using the model are compared with experimental results. The model appears to capture the physics of the upward #ame spread process, for example, the #ame tip predictions as a function of time demonstrate correct trends and often correctly predicted whether sustained #ame spread occurred. Using an experimentally measured mass loss rate in the numerical model resulted in estimates of the #ame spread rates that were lower than those experimentally measured. Several other wood-based materials, in addition to the particle board data presented, exhibited this same trend. The use of experimentally measured heat release rate data signi"cantly improved the agreement between predicted and measured #ame and pyrolysis heights but tended to overpredict the #ame spread rates. Based on this work it is suggested that heat release rate data be used in the numerical model as it results in higher #ame spread rate predictions, which is desirable in "re safety calculations. In order to obtain more reliable predictions of #ame and pyrolysis heights, a need exists for thermal property data for practical materials that will be appropriate for #ame spread models.
Acknowledgements This research was supported by the Building and Fire Research Laboratory of the National Institute of Standards and Technology under grant no. NANB8D0849.
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