Quantitative Study of Fingering Pattern Created by Smoldering Combustion

Quantitative Study of Fingering Pattern Created by Smoldering Combustion Tada Y. 1, Suzuki K. 1, Iizuka H. 1, Kuwana K. 1, *, Kushida G. 1 Yamagata University, Department of Chemistry and Chemical Engineering, Yonezawa, Yamagata, Japan Aichi Institute of Technology, Department of Mechanical Engineering, Toyota, Aichi, Japan *Corresponding author email: kuwana@yz.yamagata-u.ac.jp ABSTRACT Smoldering combustion occurs when a solid is burned in a narrow channel. A fingering pattern is then formed owing to the instability of smoldering front. We recently extended the stability analysis by Kagan and Sivashinsky to include a heat loss term and identified an ective Lewis number as the governing parameter of fingering instability. The ective Lewis number unifies the influences of material properties, experimental configuration, heat loss, and forced oxidizer flow. This study presents experimental and numerical results to further validate the stability analysis, especially, the definition of the ective Lewis number. Thin paper sheets of two different materials are burned in a narrow gap under a forced oxidizer flow. The fraction burned and the average finger width are determined by image analysis and plotted as functions of the ective Lewis number. Numerical simulations without considering heat loss or convection term are also conducted, and their results are compared with the experimental data. KEYWORDS: Fingering instability, image analysis, Lewis number, smoldering combustion. NOMENCLATURE c specific heat (J/(kg K)) D diffusivity (m /s) d channel height (m) g d s solid thickness (m) h heat transfer coicient (W/(m K)) Lewis number (-) Le ective Lewis number (-) Nu D Nusselt number (-) Q heat of combustion (J/kg) T temperature (K) t time (s) U scaled oxidizer velocity (-) u oxidizer velocity (m/s) u r reference velocity (m/s) W rate of surface reaction (kg/(m s)) Y O oxygen mass fraction (-) Greek scaled Lewis number (-) Zel dovich number (-) scaled heat transfer coicient (-) thermal conductivity (W/(m K)) density (kg/m 3 ) Subscripts g gas phase s solid phase u unburned INTRODUCTION Smoldering spread along a combustible solid often occurs at an early stage of fire. Smoldering combustion is also common in wildland fires such as those causing haze problems in Southeast Proceedings of the Eighth International Seminar on Fire and Explosion Hazards (ISFEH8), pp. 5-59 Edited by Chao J., Liu N. A., Molkov V., Sunderland P., Tamanini F. and Torero J. Published by USTC Press ISBN:978-7-31-04104-4 DOI:10.085/c.sklfs.8thISFEH.006 5

Part II Fire Asia and Northeast Europe [1]. This paper focuses on smoldering spread along a thin solid in a narrow channel, which is known to show interesting fingering instability [-4] (see also Ref. [5]). In a typical narrow channel experiment [3-6], a thin solid such as a filter paper is burned in a narrow gap between two parallel plates. A forced oxidizer flow in the opposite direction to the smoldering spread is usually supplied to avoid extinction. Previous experimental studies [3-6] showed that channel height, oxidizer velocity, and oxygen concentration in the oxidizer stream as well as material properties are key experimental parameters influencing fingering instability (see Fig. 1 for a typical experimental configuration). Among them, channel height controls the heat loss to the parallel plates as the heat transfer coicient depends on it as g Figure 1. Experimental setup. NuDλ g h =. (1) d Kagan and Sivashinsky [7] conducted a stability analysis of the fingering instability and derived a nonlinear equation for smoldering front. They numerically solved the equation and successfully reproduced a fingering pattern consisting of repetitive splits and merging. Their model, however, did not consider heat loss ect, making quantitative comparison between model predictions and experimental results difficult. We recently extended the Kagan-Sivashinsky model to include a heat loss term [8]. The dispersion relation obtained by the extended model has a single parameter, an ective Lewis number. The ective Lewis number unifies the ects of material properties, heat loss, and oxidizer flow. The results of experiments under different series of conditions supported the definition of the ective Lewis number as the governing parameter. This paper presents the results of experiments using a different paper material and newly conducted numerical simulations to further validate the theory. EFFECTIVE LEWIS NUMBER Kuwana et al. [8] based their analysis on the following -D basic equations: T t T x ( ρ gcgd g + ρ scd s s ) + ρ gcgd gu = ( λ gd g + λ gd g ) T + QW h( T Tu ) ρ YO YO gd g gd gu gd gd YO W t ρ + = x ρ, (), (3) which are obtained by the same reduction technique as Kagan and Sivashinsky [7]. Here, is the longitudinal direction, while is the transverse direction. The last term on the right hand side of Eq. () expresses the ect of heat loss. A first-order reaction of oxygen is assumed for, whereas the 53

Proceedings of the Eighth International Seminar on Fire and Explosion Hazards (ISFEH8) pyrolysis reaction is not considered (see Ref. [9] for the ects of pyrolysis reaction). Linear stability analysis to system and 3 leads to the following dispersion relation: 4 1 k 4k ω = α + InU + 1 k, U U where the scaled Lewis number, the scaled heat transfer coicient k, and the scaled oxidizer velocity are defined by the following equations: Le = 1 α k u, h, U, β = β = (5) u r where is the Zelʼdovich number, is the Lewis number defined as = + / ( ), and h is the dimensionless heat transfer coicient defined as h = h + / ( ). The reference velocity is defined as the oxidizer velocity at which smoldering spread velocity becomes zero [7-9]. Note that the wavelength is scaled by + /( ) in the present dimensionless system. With the transformation of = and =, Eq. (4) simplifies to 1 4 ˆ k ω = α InU 1 kˆ 4 kˆ + +. U (6) On the other hand, Uchida et al. [6] and Kuwana et al. [10] considered the following simple model without oxidizer flow or heat loss: (4) T T + t W, (7) Y O t 1 = YO Le W, (8) where is the ective Lewis number, and overbar denotes the dimensionless variable used in Ref. [6]. It was demonstrated that these simple equations can reproduce fingering patterns with repetitive splits and local extinction of fingertip when the ective Lewis number is sufficiently small. Linear stability analysis to system 7 and 8 yields ( α ) ω = (9) 4 1 k 4 k, where = (1 )/. Comparing Eq. (9) with (6), one can define the ective Lewis number as = +ln + / or 1 k Le Le In = U + β U Eq. (10) defines the ective Lewis number that unifies the influences of material properties, heat loss, and oxidizer flow. Eq. (6) yields the critical wavelength at which is maximal as (10) 3π l ˆ max( = Ulmax ) = β 1 1, ( Le ) (11) 54

Part II Fire which is to be compared with experimentally measured and numerically predicted finger widths. EXPERIMENTAL METHOD Narrow channel experiments similar to [3-6, 8] were conducted. Fig. 1 shows a schematic diagram of the present experiment. A thin paper sheet of 80 mm wide 300 mm long was burned in a narrow channel between two parallel plates. The same filter paper and low density paper as [6] were used, and their thermophysical properties were listed there. The vertical location of the paper sheet was at the center of the channel. A plate of heat-resistant glass was used as the top plate to enable visual observation from the top. An oxidizer flow of a specified velocity was supplied in the opposite direction to the smoldering spread. The paper edge on the downstream side of oxidizer flow was uniformly ignited using a slot burner. The following three experimental parameters were varied: the channel height,, from 7 to 1 mm; the oxidizer velocity,, from 15 to 35 mm/s; and the oxygen mass fraction in the oxidizer stream from 0.3 to 0.36. Typical fingering patterns obtained are shown in Fig.. Both finger width and fraction burned increase with an increase in channel height or oxidizer velocity. Image analysis was conducted to quantify obtained fingering patterns. An area of 80 mm 170 mm, where quasi-steady spread was achieved, was used for the image analysis. For each image, the fraction burned and the average finger width were obtained. Ten runs were conducted under each condition, and the average value over the ten runs is reported below. = 7 mm = 11 mm (a) = 5 mm/s,, = 0.3 = 15 mm/s = 30 mm/s (b) = 10 mm,, = 0.3 Figure. Experimentally obtained fingering patterns. NUMERICAL METHOD Eqs. (7) and (8) were numerically solved under varied values of. The (dimensionless) size of computational domain is 50 10. The purpose of the present numerical simulation is to compare its results with Eq. (6), obtained from more complicated Eqs. () and (3). This way, the idea of ective Lewis number introduced in this study can be tested. An explicit finite difference method was used to solve Eqs. (7) and (8). A periodic boundary condition was applied in the transverse direction. As boundary conditions on the burned and the unburned sides, zero second-derivative conditions were adopted. An initial perturbation of a sine curve was imposed, and the evolution of fingering pattern was computed. 55

Proceedings of the Eighth International Seminar on Fire and Explosion Hazards (ISFEH8) The size of computational cell was fixed at 0.05 in both transverse and longitudinal directions, and the time step was fixed at 10 6. It was confirmed that a test run with the cell size of 0.0 yielded a nearly identical result to that with the cell size of 0.05. RESULTS AND DISCUSSION Fig. 3 shows numerical results for = 0.15 and 0.4. A fingering pattern is the trace of smoldering front that has a reaction rate greater than a certain threshold value. Therefore, a fingering pattern can be numerically obtained from the distribution of the maximal reaction rate during the simulated period of time. Fingering patterns shown in Fig. 3 are thus obtained. Comparison between Fig. 3(a) and (b) shows that both finger width and fraction burned increase with an increase in. Numerically predicted fingering patterns were processed in a similar way to that described in the previous section to obtain finger width and fraction burned as functions of. 0 1.74 Temperature 0 1 Oxygen mass fraction Fingering pattern (a) Le = 0.15, t = 6 Figure 3. Numerical results. 56

Part II Fire 0 1.0874 Temperature 0 1 Oxygen mass fraction Fingering pattern (b) Le = 0.4, t = 6 Figure 3. (Continued) Fig. 4 plots experimentally measured and numerically predicted finger width as functions of the ective Lewis number,, defined by Eq. (10). The material properties and parameter values used to evaluate are listed in Ref. [8]. Eq. (11) is also shown in the same figure. As mentioned in [8], the choice of material properties changes the value of, but the conclusions of this study remain unchanged. In the experiment, as described above, finger width increases with an increase in channel height or oxidizer velocity. In the numerical simulation, on the other hand, finger width increases with. When experimental data are plotted as a function of, they are close to the numerical results, confirming that defined by Eq. (10) can unify the influences of heat loss (channel height) and oxidizer velocity. The values of under several conditions are negative, which is physically impossible. These are conditions under which the accuracy of the stability analysis is limited. Nevertheless, negative value causes no trouble in calculating finger width using Eq. (11), which agrees reasonably well with the experimental and numerical results. Experimental data under various oxygen concentrations are also plotted in Fig. 4. Under these conditions, is much less than zero, 57

Proceedings of the Eighth International Seminar on Fire and Explosion Hazards (ISFEH8) and finger width is insensitive to. More experimental data are needed to further assess the ect of oxygen concentration. Eq. (11) tends to overestimate the finger width when is less than about 0.5. A reason for the error is that the analytical model (Eq. (11)) does not consider local extinction, while both the numerical (Fig. 3) and experimental (Fig. ) results show local extinction. Fig. 5 plots experimentally measured and numerically predicted fraction burned as functions of the ective Lewis number. Unlike finger width, the fraction burned is strongly influenced by the initial disturbance, and therefore experimental error tends to be large. Overall, the fraction burned increases with an increase in. The experimental data are close to numerical results, validating the use of to combine the influences of experimental parameters. The present stability analysis considers small perturbation from the steady unperturbed spread. Therefore, the theory cannot predict local extinction and hence fraction burned. A different theoretical consideration is necessary to predict fraction burned. U (finger width) 5 0 15 10 5 Eq. (11) 0-1 0 1 Le Figure 4. Finger width and ective Lewis number. varied, = 5 mm/s,, = 0.3 (filter paper); = 10 mm, varied,, = 0.3 (filter paper); = 7 mm, = 5 mm/s,, varied (filter paper); varied, = 5 mm/s,, = 0.3 (low density paper); Numerical simulation 1 0.8 Fraction burned 0.6 0.4 0. 0-1 0 1 Le Figure 5. Fraction burned and ective Lewis number. The same symbols as Fig. 4 are used. CONCLUSIONS Narrow-gap experiments using a different paper material and numerical simulations were newly conducted to validate the prediction of previously-conducted linear stability analysis, in particular, 58

Part II Fire the definition of ective Lewis number that combines the influences of heat loss and oxidizer flow as well as material properties. The numerical model does not include convection or heat loss term, while in the experiment, oxidizer flow was supplied and heat loss to the bottom and top plates existed. Experimentally measured finger width and fraction burned were plotted as functions of the ective Lewis number, and the experimental data were in line with numerical predictions, validating its definition. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Numbers 6560171 and 15H0977. REFERENCES 1. Huang, X., and Rein, G. Smouldering Combustion of Peat in Wildfires: Inverse Modelling of the Drying and the Thermal and Oxidative Decomposition Kinetics, Combustion and Flame, 161(6): 1633-1644, 014.. Olson, S. L., Baum, H. R., and Kashiwagi, T. Finger-Like Smoldering over Thin Cellulosic Sheets in Microgravity, Proceedings of the Combustion Institute, 7(): 55-533, 1998. 3. Zik, O., and Moses, E. Fingering Instability in Solid Fuel Combustion: The Characteristic Scales of the Developed State, Proceedings of the Combustion Institute, 7(): 815-80, 1998. 4. Zik, O., and Moses, E. Fingering Instability in Combustion: An Extended View, Physical Review E, 60(1): 518-531, 1999. 5. Olson, S. L., Miller, F. J., Jahangirian, S., and Wichman, I. S. Flame Spread Over Thin Fuels in Actual and Simulated Microgravity Conditions, Combustion and Flame, 156(6): 114-16, 009. 6. Uchida, Y., Kuwana, K., and Kushida, G. Experimental Validation of Lewis Number and Convection Effects on the Smoldering Combustion of a Thin Solid in a Narrow Space, Combustion and Flame, 16(5): 1957-1963, 015. 7. Kagan, L., and Sivashinsky, G. Pattern Formation in Flame Spread over Thin Solid Fuels, Combustion Theory and Modelling, 1(): 69-81, 008. 8. Kuwana, K., Suzuki, K., Tada, Y., and Kushida, G. Effective Lewis Number of Smoldering Spread over a Thin Solid in a Narrow Channel, Proceedings of the Combustion Institute, 36(): 303-310, 017. 9. Lozinski, D., and Buckmaster, J. Quenching of Reverse Smolder, Combustion and Flame, 10(1): 87-100, 1995. 10. Kuwana, K., Kushida, G., and Uchida, Y. Lewis Number Effect on Smoldering Combustion of a Thin Solid, Combustion Science and Technology, 186(4-5): 466-474, 014. 59