Combustion Theory and Modelling Vol. 10, No. 2, April 2006, 323–347
Characterizing fingering flamelets using the logistic model S. L. OLSON∗ †, F. J. MILLER‡ and I. S. WICHMAN§ †Combustion and Reacting Systems Branch, NASA Glenn Research Center, Cleveland, OH 44135, USA ‡National Center for Space Exploration Research, Cleveland, OH 44135, USA §Mechanical Engineering Department, Michigan State University, East Lansing, MI 48824, USA (Received 24 February 2005; accepted 28 December 2005) We apply the logistic equation to a class of flame spread that occurs in near-extinction, weakly convective environments such as microgravity or vertically confined spaces. The flame under these conditions breaks into numerous ‘flamelets’ which form a Turing-type reaction–diffusion fingering pattern as they spread across the fuel. Flamelets are steady, based on flame spread measurements, and reach a critical state near extinction where a spread rate plateau reflects a critical heat flux for ignition. Our analysis of experiments performed in a buoyancy-reducing, vertically confined flow tunnel reveals the presence of statistical order in the seemingly random patterns. Flamelets as a group form a dynamic population that interacts competitively for the limited available oxygen. Flamelets bifurcate and extinguish individually, but as a whole, the group maintains a stable size. Flamelets show an exponentially decaying lifetime and a uniform pattern of dispersion. We utilize the continuous logistic model with a time lag to describe the flamelet population growth and fluctuation around a stable population characterized by the carrying capacity based on environmental limitations. We discuss how the physics of the system is expressed through the model parameters. Keywords: Near-limit; fire; flamelets; reduced buoyancy; spread rate; thermally-thin; instability; fingering; combustion; population; logistic model; bifurcation; extinction; time lag
Notation b Cs D
e f fb F Fo g h
bifurcation heat capacity of solid fuel, 1.26 J/g K binary diffusion coefficient of oxygen in nitrogen, cm2 /s = 1.6 cm2 /s at Tavg = (Tg,f + Ts )/2 = (1200 K + 400 K)/2 = 800 K; compared with D = 0.25 cm2 /s [5] extinction fractal number fraction of fuel burned number of flamelets initial number of flamelets: seven flamelets this work average gap spacing between flamelets: (w − F ∗ 2r )/(F − 1) = 0.5 cm height of test section, variable but value is 1 cm for this work
∗ Corresponding
author. E-mail:
[email protected]
Combustion Theory and Modelling c 2006 US Government ISSN: 1364-7830 (print), 1741-3559 (online) http://www.tandf.co.uk/journals DOI: 10.1080/13647830600565446
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K L m˙ f m˙ O2 m˙ O2 flamelet mw n p Pe q˙ r ro R t t t1/2 T Tign U Vf w y α γ χ O2 ρτs τ τs χ O2
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carrying capacity, number of flamelets that can be sustained in environment (12 observed in test detailed here) = m˙ O2 /m˙ O2 flamelet pyrolysis length, ∼2 mm based on experiment observations Stokes length scale (2D/U∞ ), cm fuel burning rate, gmol cellulose/s burnable oxygen supply rate, gmol/s = χ O2 pU∞ hw/(RT ) oxygen needed per flamelet, gmol/s molecular weight, g/gmol power law exponent pressure, atm Peclet number = U∞ h/D heat flux, kW/m2 flamelet radius: approximately 0.5 cm initial exponential growth rate, 0.04 bifurcations/s per flamelet ideal gas constant, 82.05 cc-atm/gmol K time, s mean lifespan, 37.2 s half-life of flamelet, 25.8 s temperature, K temperature difference between ambient and ignition temperature of the fuel, K velocity, cm/s, 5 cm/s flame or flamelet spread rate, cm/s width of the test section = 17.5 cm lateral spread width: approximately 1 cm angle, degrees lateral spread angle during bifurcation maximum change in oxygen mole fraction due to combustion 0.21 − 0.14 = 0.07 [16]. fuel area density, 7.7 × 10−3 g/cm2 time lag, s solid fuel thickness, cm equivalence ratio initial oxygen mole fraction, 0.21 (air)
Subscripts con f flamelet g gen ign normal net o O2 s unif ∞
concurrent fuel per flamelet gas per generation ignition perpendicular to the flamelet surface net heat flux, incident minus losses initial oxygen surface uniform flame free stream, ambient
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1. Introduction Flamelets are a near-limit fire adaptation to adverse conditions of limited air (oxygen) flow. These adverse conditions occur in spacecraft (in the absence of gravity) or in vertically confined spaces where there is no room for the hot gases to rise and entrain fresh air (i.e. situations where mass transport is dominated by diffusion rather than buoyant convection). In these near-limit conditions, a flame front propagating across a solid fuel breaks up into separate flamelets [1], and the propagation of the flamelets continues, leaving unburned material between the fingers of burned material (see figure 1). From a fundamental research perspective, the flamelet regime of flame spread is not well characterized. From a practical perspective, flamelet propagation constitutes a fire hazard because in this realm the flamelets survive by adapting to adverse conditions, becoming slow, weak and difficult to detect, but will likely flare up again if conditions become favourable. The patterns formed by spreading flamelets show a strong resemblance to patterns formed in biological and other physical systems. The study of pattern formation in combustion is limited [2–9]. Analysis tools with which to characterize the patterns are even scarcer. Combustion patterns, however, have much in common with patterns observed in fluids, biology, even electrochemistry [9–11]. Turing [9] was one of the first to describe these patterns, which have a characteristic length scale that spontaneously emerges in reactive–diffusive systems. His work showed that patterns arise owing to self-enhancement of random natural spatial inhomogeneities in a limiting environmental resource, resulting in the development of steep gradients in an initially uniform field. In these systems, diffusion of the limiting resource is destabilizing. Examples of Turing patterns range from the reaction fronts in catalytic mixtures (Belousov–Zhabotinsky reaction) to the stripes on zebras [11]. The complex char pattern formed by the flamelets in the test described in this paper visually resembles those found in viscous fluid fingering, dendrite seaweed tip splitting, biological systems (leaf veins, lung branching, etc.), and other branching, diffusion-controlled processes. In this system, the limiting environmental resource is the oxygen in the slow air flow. In the current paper, we apply the logistic population equation to a fingering flamelet spread test. While the detailed population analysis of only one test is presented here, we have performed
Figure 1. Video image from near the end of a horizontal fingering flame spread test, looking down on the sample with a red light illuminating the fingering pattern once flame breaks up into flamelets. First (left) third of sample was entirely consumed during flame spread prior to break-up. The copper floor of the channel shows through burned areas. Flamelets spread to the right. Flow enters from the right as shown.
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the same analysis on many tests, and the analysis described here provides meaningful trends and insights into the mechanisms driving the fingering patterns. With these interdisciplinary analysis tools we seek to improve our understanding of the physical mechanisms that can describe flamelet behaviour. 2. Experiments Experiments of flame spread across paper were conducted in a horizontal flow channel, shown schematically in figure 2, which is 30 cm wide but only a maximum of 1 cm tall. This maximum channel height is estimated to be small enough to reduce significantly buoyant recirculation,
Figure 2. Schematic of the experimental apparatus test section side and top views. The scaling of the test section is designed so that the sample is suspended in a fully developed flow between two parallel plates without the influence of side wall boundary layers.
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and consequently the constrained vertical dimension suppresses the effect of gravity on the flow field [12]. The top of the channel was made of a quartz plate to allow imaging, and the bottom of the channel was made of polished copper acting as a cold wall or heat loss surface. The WhatmanTM 44 ashless filter paper samples (area density ρτs = 7.4 × 10−3 g/cm2 ) were mounted on a frame in the centre of the channel. For the test described here, the channel height was h = 10 mm (h/2 = 5 mm on each side of the paper sheet). Air flowed both above and below the fuel sample at a controlled rate. Before and during ignition, the air flow was set to a high velocity of approximately 40 cm/s. The samples were ignited using a hot wire held in tension across the downstream edge of the sample, initiating an opposed flow flame spreading upstream. Once the flame moved away from the igniter wire, it was turned off. At a high opposed flow, the flame is continuous across the sample, burning the paper in a propagating fire line. A colour digital camcorder was used to record the test, and red light emitting diodes (LEDs) were used to illuminate the sample periodically to visualize the fingering pattern. After the igniter is turned off, the air flow was decelerated at approximately 5 cm/s2 to the desired final velocity, which in this case was approximately 5 cm/s. Within 5 s after achieving the final velocity, the flame broke into smaller flamelets, which continued to propagate in a complex pattern, as shown in figure 1. The break-up process is very dynamic, and does not always result in a uniform initial distribution of flamelets. In selected tests, the flow was reaccelerated as the flamelets approached the end of the fuel sample to determine the response of the weak flamelets to a sudden increase in air flow. For this test, the flame break-up resulted in seven initial flamelets across the sample. Once break-up into flamelets occurred, each flamelet propagated upstream, bifurcating at times to create two flamelets from the parent flamelet. The parent is considered extinguished upon bifurcation as the next generation of flamelets is started. Some flamelets extinguish without bifurcation. Each flamelet was followed through its history to record the bifurcations, extinctions, lifespan and total population as functions of time. In addition, flame spread rates and the burned fraction were measured.
3. Analysis We realized early in the analysis of this complicated system that flamelets are a near-limit adaptation of flame spread, and are stochastic in nature. Repeated tests under the same conditions gave similar but unique patterns each time. So what is a ‘steady state’ in this system? We concluded that the definition of steadiness cannot be attached to individual flamelets but must include the behaviour of the entire population of flamelets over time. Some basic expectations for a steady state are: (1) The time over which the behaviour is monitored must be significantly longer than the average flamelet lifespan, so that trends can be observed over multiple flamelet generations. A characteristic timescale of monitoring should be at least three generations. More would be desirable. On the other hand, a scenario of widely spaced non-bifurcating flamelets that continue to propagate at a constant rate has been observed (see figure 15, below, for example), so this is not a strict requirement. (2) Steady flamelets continue to propagate at an average constant rate even for very long samples. For our geometry (fixed sample width), the number of flamelets should reach a steady population that remains on average constant over timescales longer than the average flamelet lifetime. If truly steady, the bifurcation (birth) and extinction (death) rates will achieve a balance to maintain the population at a steady level.
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(3) A steady number of flamelets of a constant average size should result in a constant average fraction of the fuel sample burned. All of these average scaling quantities (lifespan, spread rate, population, bifurcation rate, extinction rate, burned fraction) can be measured from the colour video recording (29.97 frames per second) of the test. 3.1 Flame and flamelet spread rates The flame position versus time was tracked from the video recording using NASA’s Spotlight tracking software. The position–time data are shown in figure 3 for the uniform flame and three flamelets that went relatively straight upstream. Initially, the opposed flow flame linear front consumed all of the paper, and the flame spread quickly. After break-up into flamelets, the flamelet 6 spread rate was calculated, as shown by the ‘flamelet spread’ arrow in figure 1. This flamelet was chosen because it went in a fairly straight line upwind. A good linear correlation is obtained for the uniform flame spread (0.343 cm/s) and for flamelet spread (0.103 cm/s) for the time ranges indicated in the figure in parentheses. Data during transition from the flame to flamelets were not used to estimate the steady rates. While the flamelet finger propagates only 1/3 as fast as the full flame, spread persists for hundreds of seconds through numerous bifurcations. We examined the flamelet spread rates over a range of flow velocities and found an interesting trend with opposed flow, as shown in figure 4. The trend in spread rate with uniform flow is similar to what has been reported previously [13, 14] but a transition to a plateau in spread rate for flamelets is reported here for the first time. The velocity at which this plateau occurs (approximately 3.5 cm/s) agrees with the velocity observed for spacecraft flame spread transition from semicircular opposed flow flame spread to flame spread only over a limited angle in the upstream direction [13]. This plateau was found both when comparing flame spread rates taken over long tests with constant flow velocity and within single tests where the flow velocity was linearly ramped down and the spread rate was measured by differentiating the position-time data. Results for both test types are shown in figure 4. This plateau in flamelet spread rate is significant: it indicates that the flamelet is in a critical state and cannot spread more slowly or else it will extinguish. Instead of slowing down, the flamelets adjust to the decreasing oxidizer flow by shrinking and spreading further apart (via flamelet extinctions), as measured in figure 5. This figure shows one test with a linearly ramped down flow velocity where we track the burned width as a function of time. Then using a knowledge of low velocity versus time, we can then show how the total consumed fuel decreases steadily as flow decreases, despite the constant spread rate. Eventually, the flamelets become so small that the ratio of heat loss to heat generated exceeds a critical value and the flamelet extinguishes. For every material, there is a critical flux for ignition. For a thin fuel, this expression is written [15] as 1 tign
=
π q˙ net 4 ρτs Cs Tign
If we compare this with a surface energy balance for the fuel, we find it agrees except for the π /4 term with the surface energy balance if we use the concept that flame spread is a process of continuous ignition (tign = /V f ). This is shown as follows q˙ flame = ρτs Cs
∂ Ts 4 + εσ Ts4 − T∞ ∂t
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Figure 3. Flame and flamelet spread rates measured using the slope of the position versus time from the video of the test. Range of time used in the slope determination shown in parenthesis.
Figure 4. Flame and flamelet spread rates as a function of flow velocity. Data from this experiment include steady flow opposed flow spread rates (squares), linearly ramped opposed flow tests (lines), a few steady flow concurrent flow spread rates (circles), and data from a space experiment (triangles) [12]. The plateau in flamelet spread rate is correlated with the critical heat flux for ignition.
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Figure 5. Burned width versus opposed flow velocity for a linearly ramped opposed flow test, showing the linear decrease in burned area with decreased flow. This shows that the flamelet adjusts to the decreased oxidizer flux by decreasing the fuel consumption while maintaining the flamelet spread rate (figure 4).
If we approximate the temperature derivative as
Tvap − T∞ T ∼ t /V f
we can use this surface energy balance to determine this critical heat flux for ignition using the plateau value in the flame spread rate, viz Ts 4 q˙ net = ρτs Cs ∼ q˙ flame − εσ Ts4 − T∞ /V f
For property data from [13]: ρs τs = 0.0077 g/cm2 , Cs = 1.26 J/g K, Vf = 0.06 cm/s (measured plateau value), = 2 mm (see figure 14, below), and a temperature difference of approximately 700 − 300 = 400 K, we estimate q˙ net = 1.17 W/cm2 = 11.7 kW/m2 . This is within the range of critical heat flux values measured for cellulosic materials (10–13 kW/m2 ) [16].
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3.2 Flamelets are responsive to flow direction and magnitude Flamelets occur in either opposed-flow spread (flame spreading against the wind) or concurrent spread (with the wind) under weak ventilation conditions. The fingering nature of the two spread modes is different, however, as shown in figure 6, for tests at 5 cm/s with a 1 cm total channel height. The opposed flamelet rapidly tunnels through the unburned fuel, leaving well defined tracks to show the trail of the finger. The concurrent flamelet, on the other hand, is in the form of short thin flame segments shown in figure 6 that hug the upstream edge of the fuel, moving laterally back and forth along the upstream edge as shown in figure 7, consuming it over those segments, leaving much less material behind to show its trail. Since the majority of the spread is back and forth laterally, the concurrent flamelet spread rate forward is lower, as shown for a few cases in figure 4. While slower, the spread rate is still steady as shown in figure 8. In this analysis, the data are taken only when the flame is present
Figure 6. Concurrent flamelets tend to traverse the upstream edge of the unburned material, whereas opposed flamelets tend to tunnel into the pristine fuel. While the concurrent flamelets spread more slowly than the opposed flamelets, overall, they consume more of the fuel. Coordinate system shown is used in figure 7 analysis.
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Figure 7. X and Y position versus time of an edge of a concurrent flamelet, showing how the flame moves back and forth along the upstream burned edge (Y) much more quickly than it penetrates the unburned fuel (X). For the X ∼ 40 mm distance penetrated, a similar Y distance was traversed back and fourth four times.
along the tracking line parallel to the flow, and the frames where the flamelet had moved off the line were deleted (i.e. fixed position until the flamelet returned). This procedure shows that the concurrent spread rate is steady for long distances and times. If ignited at the downstream edge or in the middle of the fuel, the predominant mode is opposed flow spread, because the flame furthest upstream will consume the oxygen. Downstream reactions are unable to survive in the vitiated air [13]. However, if the opposed flow flamelet reaches the upstream edge with adjacent unburned fuel, then concurrent flamelets can survive since the fresh oxidizer reaches them directly. 3.3 Burned fraction We can approximate the amount of fuel burned for these flames by measuring the area burned compared with the total sample area. Figure 9 shows the measured fraction of sample consumed as the flamelets progressed along the sample. This was measured from the video by threshholding the black region burned (which is the copper substrate showing through) in contrast with the illuminated paper. Only minimal char remains were observed for the Whatman 44 ashless filter paper, so this is a reasonable way to estimate the amount of fuel burned. The fraction of fuel consumed increases linearly with time until the flame spread is finished and the burned fraction becomes constant, indicating burning has ended. For a linear spread rate (shown in figures 3 and 8) and constant fraction burned, a steady fuel consumption rate is expected. The fuel consumption rate is calculated from the spread rate, burned fraction and fuel area density and molecular weight as m˙ f = f b wρτs Vf /mwf . For this test, 5.1 × 10−5 gmol cellulose/s was consumed on average for the opposed flow flamelet spread, assuming complete combustion for the ashless filter paper.
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Figure 8. To estimate the concurrent flamelet spread rate, the flame tracking was constrained to a line parallel to the flow direction and frames where the flamelet was not present at that line were deleted from the data (they would have resulted in a constant position until the flamelet returned, giving a stair-step appearance to the data). The resulting spread rate is surprisingly constant, indicating this is also a steady phenomenon. The actual flamelet movement was much faster than this owing to lateral motions.
Figure 9. Comparison of the fuel fraction burned for opposed flamelets (62%) and concurrent flamelets (88%) under the same experimental conditions. The linear increase indicates the burning is steady until flamelets start to reach the end of the measurement area. While concurrent flamelets spread more slowly on average (Vf,con = 0.6 Vf,opp in this test), they consume more of the material in the long run. Time is normalized with total spread time.
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For stoichiometric burning of cellulose, 6 gmol of oxygen are needed per gmol of fuel. In this test, since the total oxygen flow was carefully controlled, we can estimate the global equivalence ratio = 6m˙ f /m˙ O2 for these flamelets. Based on ideal gas oxygen supply rate m˙ O2 = χ O2 pU∞ hw/(RT ) and fuel burning rate above, we estimate the global equivalence ratio as = 6(gmolO2 /gmolcellulose ) × (5.1 × 10−5 gmolcellulose /s)/(2.45 × 10−4 gmolO2 /s) = 1.25, which is fuel-rich. This global stoichiometric balance, which is similar to [6], shows that for these flamelets, oxygen is indeed the limiting resource that restricts the amount of sample consumed. In a few recent tests we measured the oxygen consumption at 6–7% consumed, so our assumption of 7% is reasonable. In addition, we found several percent CO in the product gases. In addition to being toxic, elevated CO concentrations are indicative of fuel-rich burning. Figure 9 also shows the differences in the fuel consumption totals for the opposed and concurrent flamelets. Both flamelets show a roughly linear increase in fuel consumed with time and distance, indicating a steady spread. While the opposed flamelets rapidly spread, consuming the fuel in a tunnelling fashion, they leave behind a significant fraction of unburned material (38%). By contrast, the concurrent flamelets burn only about half as fast, but leave only 12% of the material unburned during its lateral propogation. Thus its overall fuel consumption is actually only slightly smaller. 3.4 Population studies Flamelets show a uniform distribution pattern, as shown in figure 1. The fingers have a characteristic finger width, which scales with heat loss, and a characteristic finger separation distance, which scales with oxygen flow [5–7]. This indicates that flamelets optimize their spacing in response to environmental conditions, and they are limited by a scarce resource, which in this case is the oxygen supply. The flamelet population size is dictated first by the sample width and second by the available amount of the scarce resource, which drives the spacing between flamelets. The individual flamelet size is determined by the balance between energy generated to energy lost through conduction and radiation [3]. The bifurcations and extinctions were tracked in time from the video of the test, as shown in figure 10. The total population of flamelets as a function of time is determined from the data. An increase in population is not apparent within the scatter of the population with time (average population ∼12 ± 3). The running average population is shown for comparison. Only data after break-up are used to compute the average. Notice that the extinctions do not begin until the population approaches its average value. Each bifurcation b adds a net of one flamelet; each extinction e subtracts one flamelet, F(t) = 1+ {b(t) − e(t)}. Note that the bifurcations and extinctions show a linear cumulative number of events (i.e. d b(t)/dt = slope), which means they occur at a nearly constant rate. The 95% confidence bounds are shown for the regression analysis. The extinctions show a slightly more stair-step behaviour than the bifurcations, meaning they tend to occur in clusters or more abruptly than bifurcations, which occur rather evenly. The slopes are slightly different, but this difference is within the confidence bounds. A difference in slope suggests a gradual increase in population, but the random variations in the population exceed the changes that would be observed during the test time. The bifurcations increase substantially near the end of the test. These results are not included in the slope calculation since they may occur because of changes in the flow near the end of the sample (flow undergoes transition from fully developed duct flow to fully developed flow on both sides of the sample, see figure 1). The derivatives of the bifurcation and extinction data were examined, as shown in figures 11(a)–(c). The raw data were first smoothed using a locally weighted polynomial regression algorithm and then differentiated to determine general trends in the bifurcation and
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Figure 10. Population of flamelets from the test showing cumulative bifurcations, extinctions, and the actual population as a function of time during the test.
extinction rates. The rates show a sinusoidal trend. The oscillation around a mean value is a further indication of the dynamic nature of the stability of the population. Between 120 s and 150 s [figure 11(c)], the bifurcation and extinction oscillations line up closely, after which the bifurcations outstrip the extinctions as the population grows when the flame approaches the
Figure 11. A closer examination of the (a) bifurcation and (b) extinction rates reveals an oscillatory trend to each. The discrete raw bifurcation and extinction data were smoothed first, and then the smoothed data were differentiated in time to determine the trends with time. The smoothed derivatives, which are the bifurcation rates and extinction rates, are then fitted with waveforms as shown. There is no appreciable improvement in the fit with the rise to maximum or decaying sine forms. (c) The rates match up by the last cycle before sample end effects begin to play a role. (Continued)
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Figure 11. (Continued)
end of the sample. These oscillations in the bifurcation and extinction rates are attributed to time lags, where the two overcompensate or overreact to any change in the other. This time lag is discussed further below, as part of the logistic model. 3.5 Lifespan and generation definitions A detailed accounting of the flamelet population can be taken to determine the lifetime of each flamelet. This lifetime starts at bifurcation, and ends with either a bifurcation or extinction.
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Table 1. Flamelet demographics.
Flamelet category Bifurcated Extinguish w/o bifurcation Indeterminants (> end of test)
Number of flamelets
Percentage of total flamelets (excluding indeterminants)
32 29 9
52.5 47.5 —
There is no overlap of generations. These assumptions are important in the later use of the logistic equation, which relies on no overlap of generations and uses lifespan information in its formulation of parameters. Seventy separate flamelets were tracked. From these flamelets, three categories of flamelet types were found: (i) extinguishing flamelets, which extinguished without bifurcation; (ii) bifurcating flamelets, which bifurcated to produce the next generation of flamelets; (iii) indeterminant flamelets, which persisted to the end of the test sample, and thus could not be included in the lifespan determinations. The fraction of bifurcated flamelets that bifurcate themselves is an important measure of population stability. As shown in Table 1, slightly more than half (∼52.5%) of the flamelets bifurcated. The other remaining flamelets extinguished without bifurcation. For a population to be steady, on average, each bifurcation must result in only one viable flamelet – 50% persistence to bifurcation is necessary for steady state. We have done this same analysis on many tests, and all tests with a reasonably uniform pattern provide a similar percentage of bifurcating flamelets (50% ± 3%). Previous fingering smoulder results [4] gave similar numbers: in each case 52–53% of the smoulder spots survived to bifurcate again. In those tests, up to 588 smoulder spots were tracked. A bifurcation rate consistently slightly over 50% may indicate that rapid extinctions were not accounted for in the total. However, it might also reflect a slight growth in the populations. Since the fraction surviving is quite similar for these disparate tests, the former alternative is deemed more likely. Having determined the lifespan of each flamelet, we examine the histogram of lifespans in figure 12, which is fitted with an exponential decay model. That the lifetimes are exponentially distributed is not surprising for a stochastic branching process where the generations do not overlap. These are characteristics of a Poisson process, which is used to describe many chemical and biological processes. The average lifetime from the exponential fit is 37.2 s. The half-life from the data (ln(2) × 37.2 = 25.8 s) is about the time needed to bifurcate unimpeded (discussed below in section 3.7). Many flamelets are impeded by adjacent flamelets, hence they take longer to bifurcate. In addition, the half-life is the time required for 50% of each generation to bifurcate, and a 50% successful bifurcation rate is needed for a stable population. 3.6 Population model To describe the flamelets as a group, we make use of the equations that describe a population: dF/dt = ro F(t)(1 − F(t) / K ), which is the classic logistic equation, where the growth rate ro (1− F(t)/K ) is limited by environmental resources. Rearranging and integrating for F(0) = Fo , the solution to the classic logistic equation for a population that is limited by environmental
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Figure 12. Exponential decay fit to histogram of flamelet lifetimes for 70 flamelets during test.
resources [17] is: F(t)/K = [1 + {(K /Fo ) − 1)}e−ro t ]−1 . For the population of flamelets, the resource limiting the spacing of flamelets is oxygen supply, so the carrying capacity is the O2 supply rate divided by the O2 needed per flamelet, or K = O2 supplied/O2 required per flamelet. When the O2 needed by the flamelets equals the O2 supply, the population is at its carrying capacity. Most real populations cycle around the carrying capacity owing to intrinsic time delays in the response of bifurcation and extinction rates to environmental changes. The time delays in these tests are attributed to a spreading needed to allow the next bifurcation. The differential logistic equation with the time lag τ is: dF(t)/dt = F(t)ro [1 − F(t − τ )/K ]. Thus the population change responds to its value at some past time (t − τ ), rather than the present time t. Depending on the intrinsic (exponential) growth rate ro , and the intrinsic time lag τ , populations described by the logistic equation can exhibit one of three patterns as ro τ increases: (i) approach K and stabilize (steady), for ro τ < 1/e (ii) overshoot K followed by damped oscillations to a steady K , for 1/e < ro τ < π /2 (iii) steady oscillations (limit cycles) around K , for ro τ > π/2 The actual population is in qualitative agreement with the logistic models [18] with and without a time lag, as shown in figure 13(a). A value of ro = 0.04/s is used, since this was the initial per capita bifurcation rate estimated from the first few bifurcations (exponential growth model), in contrast to the steady population per capita growth value, which is (0.148 bifurcations/s–0.134 extinctions/s)/12 flamelets = 0.0012/s ≈ 0. As the population approaches the carrying capacity, it slows its growth, where the flamelets interact with neighbouring flamelets and some extinguish as a result of competition for the
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Figure 13. (a) Exponential and logistic models (steady and time lag) compared with actual population for opposed flow flamelets and (b) opposed and concurrent flow flamelets compared with normalized axes. Notice the overall rise time and dynamic behaviour is similar in the two cases.
limiting resource (i.e. oxygen). The same behaviour is seen for concurrent flamelets, as shown in figure 13(b). A time lag of 25 s is used, based on the observation that the actual population cycle seems to be approximately 100 s, and the period of oscillation is always approximately 4τ . This value is related to the half-life found earlier, as will be discussed below. For this experiment,
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ro τ = 0.04 × 25 = 1, which provides a damped oscillatory approach to the asymptotic carrying capacity K .
3.7 Physics behind the model parameters The logistic model with a time lag has three parameters that need to be determined: K , ro and τ . Each parameter can be defined by the important physics of the problem. Parameter K , the carrying capacity in terms of oxygen supply, is defined above as K = burnable O2 supplied/O2 required per flamelet. It can also be explicitly observed as the steady population size – in this case K = 12. This definition assumes that a steady state distribution of flamelets is established. Since the O2 supplied is a primary variable in the test, we can evaluate the burnable O2 required per flamelet. For tunnel heights comparable to flamelet radii (h/2 ≈ r ), the experimental data can be used to evaluate the burnable oxygen supplied using the ideal gas law: m˙ O2 = χ O2 pU∞ hw/(RT ), which assumes all the oxygen in the duct flows within the diffusion distance of a flamelet and is thus available for consumption. The change in oxygen concentration is defined as the initial concentration minus the fundamental limiting oxygen concentration for flammability, since a flame cannot be sustained if the ambient level falls below this value regardless of the other environmental conditions. For cellulose, this fundamental limit is assumed to be ∼14% oxygen [14]. The width w and the height h of the duct are actual dimensions (w = 17.5 cm, h = 1 cm total, p = 1 atm, T = 300 K, and the velocity U∞ = 5 cm/s). The carrying capacity can thus be expressed as K = m˙ O2 /m˙ O2 flamelet . For this test, 2.45 × 10−4 gmol O2 /s were supplied. For an observed steady population of K = 12 flamelets, each flamelet is allotted 2 × 10−5 gmol O2 /s flamelet (∼0.5 cc/s O2 , or ∼2.4 cc/s air). Our tests measuring the amount of oxygen consumed have not found less than 14% oxygen remaining, so our burnable oxygen estimate above is reasonable. The above quantity of oxygen needed per flamelet has been found to be reasonably constant over a range of conditions that result in flamelet populations ranging from 3 to 12. The population rises as U∞ × h increases in air, as expected. Since the primary mode of oxygen transport between flamelets is diffusion, as the supply of burnable oxygen is reduced, the spacing between flamelets increased, which is consistent with [4–6]. As used earlier, a time lag τ of 25 s seems to describe the actual population cycle of approximately 100 s, since the period of oscillation is 4 τ . The time lag is best associated with the time it takes one flamelet to affect the bifurcation potential of the others around it. The lag is representative of the developmental period, or time to spread enough for bifurcation or the bifurcation time itself. For flamelets, the lag is attributed to the time for lateral spread rate of the flamelets, which affects the bifurcation rate as well as encroachment effects (i.e. extinction) on adjacent flamelets. For example, when a flamelet extinguishes, the adjacent flamelet(s) begin to spread into the vacated flow area, but it takes time to spread laterally enough to bifurcate. In order to bifurcate, each flamelet must spread laterally to roughly twice its normal width (2r ∼ 1 cm), as shown in figure 14. The lateral spread distance (y ∼ 2r ) depends on the spread rate, time, and the angle of lateral growth as y = V f t tan γ . As seen in figures 2 and 14, the lateral growth angle γ is generally 15–23◦ , with an upstream spread rate of ∼0.1 cm/s (figure 3). We can estimate the lag as τ ≈ 2r/(V f tan γ ), which ranges from 24– 37 s for our observed range of angles, in good agreement with the 25 s lag estimated from the period of oscillation. In addition, these values are between the half-life and lifespan of flamelets calculated above. Bifurcations will then occur every 2.4–3.7 cm along the finger. For a steady population at the carrying capacity, a flamelet must extinguish for each flamelet that
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Figure 14. Flamelet bifurcation sequence, showing lateral growth followed by extinction of the lagging central flame zone. Bifurcation angle between two flamelets is ∼ 46◦ , or 23◦ for each from the direction of spread.
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bifurcates. Thus, the half-life from the census data may be an excellent way to estimate the lag time. The angle of lateral growth has been shown to be a direct function of the flow velocity normal to the flame front [13]. As the flamelet front widens, the central portion of the flame weakens as it becomes further removed from the edges, and thus receives less oxygen. There is a narrow range of conditions where the flame cannot sustain itself without lateral oxygen diffusion from the edges, but can sustain itself with the augmention of the edge diffusion. It is in this window of conditions that we find the fingering flamelets. The flamelet curves into cusps as the central portion falls behind the enhanced edges. The normal angle relative to the flow increases and the trailing cusp tips receive less oxygen from the flow (Unormal = U∞ cosγ , although lateral diffusion augments the forced flow of oxygen on the outside edges). It is not hard to envision a sufficiently low oxygen supply where the flamelet cannot spread far enough laterally to bifurcate. This type of tunnelling flamelet has been observed in our tests, as shown in figure 15, as well as in [4–6, 13], where the undulating char pattern indicates successive failed attempts to bifurcate. The intrinsic per capita growth rate, ro , is the exponential growth rate without resource limits. To determine an appropriate scaling for ro , we look to other systems with similar pattern formation – in this case tip splitting in directional solidification. The tip splitting frequency (bifurcation rate) was related to the growth rate (spread rate) via a power law relation [19]. Power laws are ubiquitous in the allometric scaling literature [11, 20], and are characteristic of fractal patterns. For many systems, the exponent n is 3/4 based on energy minimization. Fingering smoulder experiments found n ∼ 0.6 [4]. The fractal numbers f of the fingering patterns in our tests has been measured using the box-counting method and are typically ∼ f = 1.7 ± 0.1. An unbroken flame front has a fractal number of 2 (2D), whereas in the thin limit, a single, straight, non-bifurcating finger would have a fractal number of unity (1D line). Using the power law formulation, we express the intrinsic per capita growth rate as ro ∝ ( f −1) (V f ×t L /L)n , where L is a characteristic length scale in cm, t L is the average lifespan of 37.2 s, and the ( f −1) factor is the fractal number such that as ro → 0 (bifurcation limit) as f → 1 (line) for the observed sub-bifurcation limit tunnelling spread (finite flow and spread rate). The characteristic length scale can be chosen as the diffusive gas-phase Stokes length scale L = 2D/U∞ . For this experiment L = 0.64 cm, which is qualitative agreement with the average gap spacing between flamelets g ∼ (w − 12 ∗ 2r )/(12 − 1) ∼ 0.5 cm. In addition, the half-duct height (h/2) = 0.5 cm is in close agreement with the flamelet radius ∼0.5 cm. Thus ro ∝ ( f − 1)(U∞ V f t L /2D)n . These relations agree with [4–6], which showed that flamelet size correlated with heat loss (∝ 1/ h), and the spacing was shown to correlate with oxygen flow. As the oxygen flow becomes more limited, the diffusive length scale increases and the intrinsic per capita growth rate decreases until no more growth (bifurcation) is possible, as shown in figure 15. These scaling estimates agree with observed trends: (i) flamelets form as the convective flow approaches diffusive flows (Stokes length scale L → 1 cm); (ii) the steady population (K ) increases with increasing U∞ , consistent with [4–6]; (iii) flamelets merge into a single flame at sufficiently high flow (gap ≈ L → small as U∞ → large); (iv) the intrinsic population growth rate (ro ) increases as a power law function with U∞ , consistent with [4] whose data exhibited a power law relation with an exponent n ∼ 0.6; (v) ro → 0 (bifurcation limit) as U∞ → 0 and/or f → 1.
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Figure 15. After flow ramp down, only three flamelets survived, with the rest of the line extinguishing. They formed a fingering pattern without viable bifurcation. Despite repeated attempts to grow laterally (note undulating char pattern), no viable bifurcations were possible. A similar pattern was seen for the weakest flows on the Space Shuttle [13]. Conditions were 3 mm spacing on either side and 5 cm/s final flow.
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Figure 16. Weak flamelets flare up as flow increases. The flamelets grow and expands from (a) the small flamelets at 3 cm/s flow to (b) a merged wavy blue front with luminous sooty tails at each flamelet at 2.5 s, and finally (c) after only 6 s into a full luminous front at 33 cm/s (flow increased from 3 cm/s to 33 cm/s).
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3.8 Fire safety implications Flamelets are by nature small and hard to detect, and occur at the limits of extinction. As demonstrated in these tests, they occur in confined spaces that mimic wall or ceiling cavities in buildings or on aircraft. Flamelets can persist indefinitely under the right conditions, and can flare up again into a large fire if the environmental conditions change. For example, in figure 16 a case of fingering flamelets is shown. Then the flow was suddenly increased from 3 cm/s to 33 cm/s, the flamelets rapidly expanded radially, remerged into a flame front, and began to spread much faster (see figure 4). At sufficient flowrates (∼>15 cm/s), the concurrent flame receives enough oxygen to become simultaneously viable [21], and a downstream flame begins to consume the unburned material left behind by the fingering flamelet spread. A small flame in a confined space, if undetected, can lead to devastating consequences. A fire in the cockpit ceiling of Swissair Flight 111 in 1998 ignited and spread undetected, and ultimately doomed the aircraft and passengers. A detailed description of the fire was given by the Transportation Safety Board (TSB) of Canada, which oversaw the recovery and reconstruction of the aircraft to study the cause of the accident [22]. The TSB concluded that arcing wires ignited insulation and a creeping flame ensued that slowly spread in a very weak ventilation flow until air conditioning duct end caps failed and allowed fresh air into the area. The first sign of trouble was an abnormal odour in the cockpit, then a small amount of smoke became visible in the cockpit. However, the smoke then stopped entering the cockpit for a time. The crew diverted to Halifax, Canada, and during the landing preparations they were unaware that the fire was rapidly spreading above the cockpit. About 13 min after first signs of trouble, the aircraft began to suffer a rapid succession of systems failures. Only then did the crew declare an emergency. A minute later, all communication with the aircraft was lost. During the investigation, it was revealed that seven other similar incidents with fire in aircraft insulation had occurred, although none ended with fatalities. It is standard practice on US spacecraft for the astronaut crew to turn off the ventilation to help with the extinguishment of a fire, both to eliminate the fresh oxygen supply and to reduce the distribution of the smoke. A thorough understanding of flamelet character is needed to help astronauts detect and fight fires, and ensure complete extinguishment. If the crew thinks the fire is fully extinguished, but some dim, blue, tiny flamelets go undetected, the crew might then reactivate the ventilation system to clear the smoke. The flamelets could grow into a large fire very rapidly. A similar event has already happened aboard the Russian Mir Space Station. On 15 October 1994, the filter of the solid fuel oxygen generator caught fire (not the infamous 23 February 1997 fire, which was much worse). Cosmonaut Valery Polyakov was the first to reach the unit and put the fire out using a jumpsuit belonging to crewmate Yuri Malenchenko – the first thing he could find. Shortly after extinguishing the fire, however, the cosmonauts discovered that the jumpsuit was smouldering and had burned a hole through the chest area. Had the crew not noticed this smouldering fire, it could have grown undetected into a large damaging fire.
4. Conclusions We have shown through the novel application of bio-mathematical tools, that flamelets are a stable flame phenomenon in an aggregate sense. Our analysis of a fingering flamelet spread experiment has revealed the presence of mathematical statistical order in a seemingly random system. Flamelets form a dynamic population that interacts competitively for limited available oxygen resources. Flamelets show many of the same characteristics found in plant and animal populations, hence many of the bio-mathematical concepts used to describe biological
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populations can be used to describe flamelets. As a population, flamelets exhibit a Poisson process with an exponential lifespan histogram and a uniform pattern of dispersion. We utilize the continuous logistic model with a time lag to describe the behaviour of the flamelet population after break-up. Flamelets reproduce through bifurcation (tip-splitting) and die either at bifurcation (when the next generation begins) or through extinction without bifurcation. The population interacts competitively around a dynamic equilibrium population size at a carrying capacity defined by the supply rate of oxygen. The value of the time lag τ is estimated by considering the time needed for a flamelet to spread sufficiently far laterally that two selfsufficient flamelets can form. A low-velocity bifurcation limit is associated with a limiting velocity (oxygen supply) normal to the curved flamelet surface. Lastly, the intrinsic bifurcation rate is estimated as a power law function of the spread rate. Enhanced understanding of fingering flame spread may be relevant to spacecraft fire safety, or to fires in confined spaces such as wall or ceiling cavities in buildings or on aircraft.
Acknowledgements The authors gratefully acknowledge the contributions of MSU graduate students Lisa OraveczSimpkins and Stefanus Tanaya who designed the Hele-Shaw facility and conducted the experiments that were discussed herein, and Karin Aditjandra, who conducted similar analyses on numerous other tests. The authors also are indebted to the ATHINA project team, who provided engineering support for the work. This research was supported by the NASA Office of Biological and Physical Research Cooperative Agreement No. NCC3-1053 to Michigan State University, and through internal funding at NASA Glenn Research Center at Lewis Field. References [1] Lewis, B. and von Elbe, G., 1987, Combustion, Flames and Explosions of Gases, 3rd edition, pp. 326–327 (Academic Press: New York). [2] Olson, S.L. and T’ien, J.S., 2000, Buoyant low stretch diffusion flames beneath cylindrical PMMA samples.Combustion and Flame, 121, 439–452. [3] Olson, S.L., Baum, H.R. and Kashiwagi, T., 1998, Finger-like smoldering over thin cellulosic sheets in microgravity. Proceedings of the Combustion Institute, 27, 2525–2533. [4] Zik, O., Olami, Z. and Moses, E., 1998, Fingering instability in combustion. Physical Review Letters, 81, 3868–3871. [5] Zik, O. and Moses, E., 1998, fingering instability in solid fuel combustion: the characteristic scales of the developed state. Proceedings of the Combustion Institute, 27, 2815–2820. [6] Zik, O. and Moses, E., 1999, Fingering instability in combustion: an extended view. Physical Review E, 60, 518–531. [7] Aldushin, A.P. and Matkowsky B.J., 1998, Instabilties, fingering, and the Saffman-Taylor problem in filtration combustion. Combustion Science and Technology, 133, 293–341. [8] Nayagam, V. and Williams, F.A., 2000, Rotating spiral edge flames in von Karman swirling flows. Physical Review Letters, 84, 479–482. [9] Turing, A.M., 1952, The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society B, 237, 37–72. [10] Godreche, C. (editor), 1992, Solids far from Equilibrium (Cambridge: Cambridge University Press). [11] Ball, P., 1999, The Self-Made Tapestry: Pattern Formation in Nature (Oxford: Oxford University Press). [12] Oravecz, L.M., 2001, Instabilities of Spreading Diffusion Flames in Microgravity and the Design and Construction of a Hele-Shaw Apparatus That Produces Flames in the Near Extinction Limit Regime Under Simulated Low Gravity Conditions. MS Thesis, Michigan State University, Department of Mechanical Engineering. [13] Olson, S.L., Kashiwagi, T., Fujita, O., Kikuchi, M. and Ito, K., 2001, Experimental observations of spot radiative ignition and subsequent three-dimensional flame spread over thin cellulose fuels. Combustion and Flame, 125, 852–886. [14] Olson, S.L., 1991, Mechanisms of microgravity flame spread over a thin solid fuel: oxygen and opposed flow effects. Combustion Science and Technology, 76, 233–249. [15] Tewarson, A., 1995, Fire properties of materials. Improved Fire- and Smoke-Resistant Materials for Commercial Aircraft Interiors: a Proceedings (Washington: National Academy of Sciences), pp. 61–91.
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[16] White, R.H., and Dietenberger, M.A., 2001, Wood products: thermal degradation and fire. Encyclopedia of Materials: Science and Technology [S.l.] (Elsevier Science Ltd), pp. 9712–9716. [17] Williams, B.K., Nichols, J.D. and Conroy, M.J., 2002, Analysis and Management of Animal Populations (Boston: Academic Press). [18] Alsted, D., Populus, Java Version 5.2.1, University of Minnesota, March 2003. Available online at: www.cbs.umn.edu/populus/. [19] Utter, B., Ragnarsson, R. and Bodenschatz, E., 2001, Alternating tip splitting in directional solidification. Physical Review Letters, 86, 4604–4607. [20] West, G.B., Brown, J.H. and Enquist, B.J., 1999, The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science, 284, 1677–1679. [21] Prasad, K., Olson, S.L., Nakamura, Y. and Kashiwagi, T., 2002, Effect of wind velocity on flame spread in microgravity. Proceedings of the Combustion Institute, 29, 2553–2560. [22] Transportation Safety Board of Canada Report Number A98H0003, 1998, Aviation investigation report: inflight fire leading to collision with water.
