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Determination of the spread rate in opposed-flow flame spread over thick solid fuels in the thermal regime
Twenty-Sixth Symposium (International) on Combustion/The Combustion Institute, 1996/pp. 1477–1485
DETERMINATION OF THE SPREAD RATE IN OPPOSED-FLOW FLAME SPREAD OVER THICK SOLID FUELS IN THE THERMAL REGIME SUBRATA BHATTACHARJEE and JEFF WEST Department of Mechanical Engineering San Diego State University San Diego, CA 92182, USA ROBERT A. ALTENKIRCH School of Mechanical and Materials Engineering Washington State University Pullman, WA 99164, USA
Limitations of the assumptions in the classic de Ris formula [1] for opposed-flow flame spread rate over thick fuels in the thermal regime are explored through numerical experiments in which the assumptions used in the original theory are eliminated one by one from the mathematical model. While the assumptions of infinite-rate chemistry, mass flux linearization, and neglect of radiation are found to be reasonable in the thermal regime, assumptions of a slug flow, a surface-hugging flame, and constant properties are responsible for the predictions of spread rate being in error by as much as a factor of 10. A spread rate formula, derived from a semiempirical solution of a simplified problem, is shown to overcome the drawbacks of the de Ris formula while retaining its simplicity. A hydrodynamic coefficient, developed for two different flow configurations, and a flame liftoff coefficient, both obtainable from the known parameters of the problem, correct for some of the major limiting assumptions in the de Ris theory. The formula developed here, which can be written as 2
FEST ˆ Vˆ f,EST 4 Veqv b5b6 in which Vˆeqv contains the hydrodynamic information and the factor in front of it the chemical information, performs significantly better than the de Ris formula when tested against available data and extensive computational results for different flow configurations, opposed-flow velocities, oxygen levels, and ambient pressures.
Introduction Flame spread over a thermally thick fuel can be classified into three distinct regimes based on the strength of the opposing flow (1) the low-velocity quenching regime in which radiative effects are important [2,3], (2) the high-velocity blow-off regime in which chemical kinetic effects are important [4], and (3) the moderate-velocity thermal regime [1,5] in which both the radiative and chemical kinetic effects are relatively unimportant. Of these, only the thermal regime, because of its simplified physics, has yielded useful, closed-form solutions to model problems for the spread rate. The classic de Ris formula for the spread rate in the thermal regime, based on a number of symplifying assumptions, is probably the most well-known formula in the opposed-flow flame spread literature [6]. It has found use beyond characterization [4] of the thermal regime in which by comparison of the
spread rate with the de Ris formula attempts have been made [4,7] to identify the particular physics that delineate new regimes. The Damko¨hler number correlations of Fernandez-Pello et al. [4] and Altenkirch et al. [7] are some cases in point. To date, only a handful of studies have attempted to assess the impact of the various assumptions on the spread rate formula, the most notable being the work of Wichman [5], who showed that removal of the Oseen approximation can cause a significant difference in the flame spread rate. The Wichman formula, however, contains two undetermined constants and a velocity gradient that is not readily known. Here, we undertake a systematic evaluation through numerical case studies of the effects of the entire set of assumptions used by de Ris. We identify three major drawbacks of the de Ris theory that can cause large errors (a few hundred percent) in the prediction of spread rate. A simplified theory is
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Fig. 1. Computed and predicted (EST) spread rates for the baseline case (50/50 O2/N2 by volume, PMMA, 40 cm/ s, FDC) corresponding to different models with the physics added for each model change as follows: Model 1, de Ris model (zero hang distance); Model 2, nonzero hang distance; Model 3, Vˆrelative 4 Vˆg ` Vˆf ; Model 4, non-Oseen flow (i.e., x momentum equation solved); Model 5, wall blowing (i.e., y momentum equation solved); Model 6, variable gas density; Model 7, variable transport properties; Model 8, finite-rate combustion kinetics; Model 9, finiterate pyrolysis kinetics; Models 10–13, models with reradiation, gas radiation, and radiation feedback; Model 14, experiment from Ref. 4.
developed, and a semi-empirical, modified spread rate expression is proposed that retains the elegant simplicity of the de Ris formula yet corrects for its severe assumptions. Two flow configurations are investigated: (1) Fully developed channel (FDC) flow with the channel height much larger than the flame standoff distance, that is, the centerline velocity is assumed to remain unaffected by the spreading flame, and (2) forced flow over a flat plate (FP). Mathematical Models The mathematical models are listed in Fig. 1, with Model 1 corresponding to the classic de Ris formula and Model 14 representing the experimental data of Fernandez-Pello et al. [4] obtained in the FDC configuration. The spread rates in Fig. 1 are for a baseline case, that is, spread over a thick slab of PMMA flush mounted on the wall of a two-dimensional channel with an opposing, fully developed flow of a 50/50 O2/N2 mixture, by volume, with a centerline velocity of 40 cm/s at 1 atm pressure. These conditions are chosen to duplicate experimental conditions for comparison and to ensure that the case belongs to the thermal regime. In the numerical solution [8] (Model 2) to the de Ris problem [1], the flame front is not assumed to coincide with the pyrolysis front. The major assump-
tions are gradually removed in the subsequent models: (1) in Models 3, 4, and 5, hydrodynamic effects are included, (2) in Models 6 and 7, hot-gas expansion and variable properties are introduced, (3) in Models 8 and 9, finite-rate gas-phase and solid-phase chemistry are introduced, and (4) in Models 10 through 13, surface and gas-phase radiation are introduced. Steady-state equations solved in the most comprehensive model (Model 13) include the energy equation in the solid phase, and continuity, x- and y- momentum, species equations for fuel, oxygen, and nitrogen, and energy in the gas phase, the ideal gas equation and a square root dependence of transport properties on temperature. Major assumptions that remain in Model 13 include unit Lewis number, constant molecular weight and specific heats, single-step chemistry for combustion and pyrolysis, constant solid density, no surface regression, and no soot production. The submodels used in this model [9] comprise Lengelle’s pyrolysis model [2,10], Arrhenius second-order kinetics for the gas phase, and the radiation model of Bhattacharjee and Altenkirch [9]. An iterative scheme [9] in which the gas- and the solid-phase equations are alternately solved using the SIMPLER algorithm is used for the numerical solution of the spread rate and the conserved variables. Details of the mathematical formulation, numerical solution procedure, and code validation can be found elsewhere [9,11] and are omitted here for brevity. On the very left of the abscissa in Fig. 1 is the exact solution of de Ris, and on the extreme right is the experimental result of Fernandez-Pello et al. [4], with the intermediate points being the computational solutions with the assumptions of de Ris removed one by one as listed in Fig. 1. In evaluating the spread rate with the de Ris formula, the properties are evaluated at the ambient temperature, and the linearized, adiabatic temperature is used for the flame temperature. The hang distance, the distance between the flame leading edge and the pyrolysis front, is postulated to be zero in the de Ris formulation (Model 1) [11], although the underlying mathematical model allows for the fuel to diffuse upstream to form a finite hang distance. For a thin fuel, this inconsistency in the “exact” solution [11] was found to have a significant effect on Vˆf. However, for the thick fuel, with the flame computed to be on the surface as de Ris had postulated, the hang distance is found to be zero in the numerical solution, and the exact solution is reproduced (within 0.01%) in Model 2. In Model 3, the velocity the flame encounters is modified from Vˆg, used in the de Ris model (Model 1), to Vˆrel 4 Vˆg ` Vˆf, in flame-fixed coordinates. The spread rate does not change significantly because Vˆg k Vˆf in the thermal regime. A large change in Vˆf occurs when the momentum equations are
SPREAD RATE IN OPPOSED-FLOW
solved (Model 4) because the effective opposing velocity near the flame leading edge is reduced drastically by the formation of the boundary layer. Removal of mass transfer linearization by including surface blowing (Model 5) does not change Vˆf appreciably. Variable gas density and thermal conductivity (Models 6 and 7) have significant effects. In these models, the ideal gas equation is used to compute local density while a square root dependence on temperature is used to evaluate the thermal conductivity and viscosity. Finite-rate combustion chemistry (Model 8), for the kinetic constants used here, has no effect in the thermal regime. However, pyrolysis kinetics (Model 9) have a significant effect on Tˆv and hence the spread rate. Different radiation models [9], surface reradiation (Model 10), gas radiation loss (Model 11), gas and surface radiation loss (Model 12), and gas and surface radiation with radiation feedback from the gas to the solid (Model 13) affect the flame size and structure significantly. However, the spread rate appears to remain largely unaffected. Numerical results also establish that the dependence of Vˆf on the opposing flow velocity, Vˆg, deviates significantly from the linear relationship of the de Ris formula. With the momentum equations included (Model 4), the dependence of Vˆf on Vˆg for the FP and the FDC flow configurations are quite different, which is missed by the de Ris formula. The Wichman formula accounts for this geometrical dependence through a velocity gradient term, which is difficult to evaluate without experimentally determined velocity profiles. A major drawback of the de Ris theory stems from its prediction of a surface-hugging flame [8,12] that releases just enough heat to sustain a flame temperature equal to the vaporization temperature. Once the Oseen approximation is removed (Model 4), the flame lifts from the surface, and the flame temperature rapidly increases in the downstream direction. Flame liftoff is accompanied by a reduction in the spread rate, which is a different effect from the hydrodynamic effects discussed later. Finally, the existing spread rate formulas suffer from inconsistent use of three characteristic temperatures, the linearized adiabatic flame temperature, Tˆf,ad,lin, and the reference temperatures for evaluation of gas density and thermal conductivity, Tˆref,q and Tˆref,k, respectively. Use of these temperatures varies widely, resulting in disparate prediction of spread rate for the same case. For example, for the same baseline case of Fig. 1, the spread rates from the de Ris formula from different sources are: 14.5 mm/s (Eq. [6]), 6.4 mm/s [12], and 1.59 mm/s [4]. The constant property results of Fig. 1 were obtained with Tˆref,q 4 Tˆref,k 4 Tˆ`. Any modified formula must clearly specify these temperatures.
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Simplified and Extended Simplified Theory (ST and EST) As a guide in developing the EST spread rate formula, it is instructive to revisit the de Ris model and a simplified solution [11] for the spread rate (ST). The governing equations for the Schvab–Zeldovich conserved scalars, the solid-phase temperature, and the energy balance at the vaporizing interface, with mass-transfer linearization as used by de Ris [1], can be written in terms of six known parameters, b1 . . . b6, which are, in order, yo,` , s
Dhˆ oc Tˆ v , , cˆgTˆ ` Tˆ `
|Dhˆ ov| kˆ s qˆ sCˆ s , , cˆgTˆ ` kˆ g qˆ gCˆ g
To obtain an approximate solution in the downstream region, the boundary conditions are simplified by assuming that the flame lies along the vaporizing surface so that yF 4 yo 4 0 there. In addition, the governing differential equations are made parabolic, resulting in the following solutions for the coupling functions in the downstream region from the flame leading edge:
f1,ST b 4 4 f2,ST 4 erfc(gST) 1B b1b2 f3,ST 4
b3 1 1 erfc 1gST b4
1
where gST [
!
2
bthick,ST b5
yˆ ˆ 2!Lg,ST(xˆeig 1 xˆ)
(1)
Upon substitution of these expressions into the interface energy balance, an expression for the dimensionless spread rate is obtained, which is identical to the de Ris formula: bthick,ST 4
F2ST b5
where FST [
Tˆ f,ad,ST 1 Tˆ v Tf,ad,ST 1 b3 ˆTv 1 Tˆ ` 4 b3 1 1
Tf,ad,ST 4 1 ` b1(b2 1 Kb4)
(2)
Here, Tˆf,ad,ST is the asymptotic, adiabatic, stoichiometric flame temperature with mass flux linearization that appears in the de Ris formula. The simplified solution to the de Ris model, although valid only in the downstream region, provides insight into the limitations of the model and, when compared with numerical solution of more advanced models, leads to the corrections developed here. Lifted Flame and Hydrodynamic Corrections From the computed solution of the baseline case for four different models (Models 2–5) of Fig. 1, the
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here, Vˆg is replaced with an equivalent velocity, Vˆeqv 4 chydVˆg, where chyd, a second unknown coefficient at this point, is to contain the information about the character of the opposing flow at the flame leading edge. With introduction of clift and chyd, the problem admits a solution, the EST, of the same form as the ST with Lˆg,EST [ aˆ g/chydVˆg f1,EST b4 4 f2,EST 4 erfc(gEST) 1B cliftb1b2 f3,EST 4
Fig. 2. Computed surface profile of the second coupling function for different models for the baseline case. Models 4 and 5 approach their asymptotic values in the downstream direction.
interfacial value of the coupling function f2, normalized by its value at the flame, f2,flame 4 b1 b2/b4 4 7.65, is plotted in Fig. 2. As expected for the de Ris model (Model 2), f2/f2,flame assumes a constant value for x , 0 as the flame lies along the vaporizing surface. The correction of the velocity boundary condition (Model 3) does not seem to have any effect on this profile. However, as the slug flow approximation is replaced with a fully developed velocity profile in Model 4, and then the mass transfer linearization at the interface is replaced with the proper blowing term in Model 5, the f2 profiles change dramatically. Ahead of the flame (x . 0 in Fig. 2), f2 decreases more gently, as the distance over which species diffuse ('aˆ g/Vˆg) increases because of a lower opposing velocity near the wall. Downstream f2 increases sharply near the leading edge and continues to increase gently toward the corresponding asymptotic value [13] of ln(1 ` B) f2,flame (4 7.6f2,flame) b1 for Model 4 and B(1 ` b1) f (4 4.0f2,flame) b1(1 ` B) 2,flame for Model 5. The flame is lifted in both Model 4 and Model 5. To account for this in the EST, we write f2,wall,EST 4 cliftf2,flame, where the coefficient clift, an unknown at this point, is to be correlated with known parameters. The Oseen flow approximation (i.e., the assumption of a slug flow) is responsible for the proportionality between Vˆf and Vˆg, which is seen to be absorbed in the definition of bthick in the de Ris formula (bthick,ST 4 F2ST/b5). In the modified formulation
where gEST [
b3 1 1 b4
1
erfc 1gEST
!
2
bthick,EST b5
yˆ
(3)
2!Lˆ g,EST(xˆeig 1 xˆ)
bthick,EST 4 chyd
F2EST b5
where FEST [
Tf,ad,EST 1 b3 b3 1 1
Tf,ad,EST 4 1 ` b1(b2 1 cliftKb4) (4) The preceding reduces to the de Ris solution for chyd 4 clift 4 1. However, a number of favorable consequences results from a clift . 1 and chyd , 1. First, the spread rate is reduced compared to the de Ris prediction, as can be verified by comparing Eqs. (4) and (2). Second, the length scale is enlarged (Lˆg,EST 4 Lˆg,ST/chyd), which is consistent with the enlargement of the preheat zone for Model 4 in Fig. 2. Third, the flame is lifted, and the flame location can be found from Eq. (3) to be the locus of erfc(gEST,flame) 4 1/clift. The flame temperature can be obtained from the solution for the first coupling function f1,EST:
1
2
1 (5) clift For clift 4 1, Tf,EST reduces to the vaporization temperature b3 consistent with ST and the de Ris model. Note that it is the adiabatic flame temperature, Tˆf,ad,EST, and not the actual flame temperature, that appears in the spread rate formula, Eq. (4). Tf,EST 4
b3
clift
` (1 ` b1b2) 1 1
Evaluation of the Coefficients chyd and clift To develop a functional relation for chyd, attention is focused on the velocity profile encountered by the flame leading edge submerged in a boundary layer. Because the length scale there is Lˆg,EST, the proper scale for Vˆeqv is postulated to be proportional to the velocity at a distance Lˆg,EST from the fuel surface at the leading edge [14]. Depending on the flow configuration of interest, Vˆeqv and, hence, chyd, can be written as
SPREAD RATE IN OPPOSED-FLOW
Fig. 3. Computations and the EST prediction for various mathematical models for the FDC configuration.
aˆ g 1.5Vˆ g ˆ FDC: Vˆ eqv ; ˆ Lg,EST ⇒ chyd,FDC ; ˆ ˆ (H/2) HVg
1/2
1 2
aˆ 4 ch,FDC ˆ ˆg HVg
1/2
1 2
1/2 Pr11/2 4 ch,FDC Re1 H
(6)
aˆ g Vˆ g ˆ FP: Vˆ eqv ; Lg,EST ⇒ chyd,FP ; dˆ xˆdVˆ gPr
1
4 ch,FP
aˆ g
1/4
1xˆ Vˆ Pr2 d g
1/4
2
1/4 Pr11/2 4 ch,FP Re1 xd
(7) Although the hydrodynamic constants, ch,FDC and ch,FP, remain to be determined, the usefulness of these expressions is evident when substituted into the spread rate formula, Eq. (4). The dependence of Vˆf on Vˆg is correctly predicted by this modified equation. To obtain a functional relationship between clift and the governing parameters of the EST, note that in the far downstream region the asymptotic solution corresponding to Model 4 yields a value of ln(1 ` B)/b1 for clift. Therefore, clift can be expected to depend on this parameter. A selected set of computations, covering a wide range of the thermal regime, both for the channel flow and the flat plate configurations, were carried out using Model 4 (i.e., the model that includes the momentum equations without wall blowing). By fitting the flame location predicted by the EST, clift can be calculated for the baseline case. The entire data set indicates a linear relationship independent of the flow configuration and environmental conditions. A least-square-error fit yields: clift 4 1 ` clift
B(1 ` b1) , clift 4 0.09 b1(1 ` B)
(8)
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By matching the computed and predicted spread rates, ch,FDC and ch,FP are obtained and are found to be almost constants as expected. The best-fit values are ch,FDC 4 7.65 for the channel flow and ch,FP 4 2.65 for the flat plate geometry. The flat plate value is understandably smaller because the boundary layer thickness for this geometry is significantly underestimated by scaling whereas for the fully developed channel flow it is known exactly. Computations of flame shapes show that blowing does not appear to have much impact, especially in the FDC configuration; the same is true for the spread rate as shown in Fig. 1. The EST developed for Model 4, then, is expected to work for Model 5. To incorporate the variable density effect (Model 6), which results in a significant decrease in Vˆf without much effect on the slope of Vˆf versus Vˆg as can be seen in Fig. 3, the gas density in the EST is evaluated at Tˆref,q 4 Tˆv. Good agreement is obtained between the prediction and Model 6 computations for the baseline case of Fig. 1 and the test cases of Fig. 3. When thermal conductivity is allowed to vary with the square root of the local temperature (Model 7), Vˆf increases sharply as shown in Figs. 1 and 3. This increase is found to be independent of the ambient oxygen level. Therefore, once again, a reference temperature Tˆref,k 4 Tˆv is chosen for the evaluation of thermal conductivity. The resulting modified formula (EST Model 7), although an improvement, still underpredicts (Fig. 1) the spread rate. It is not surprising that the vaporization temperature, Tˆv, turns out to be a reasonable temperature for the evaluation of properties, as opposed to the adiabatic, stoichiometric flame temperatures based on complete or equilibrium combustion [4,5,15]. Near the flame front, the flame temperature rapidly decreases as the leading edge is approached, and it is almost equal to Tˆv at the leading edge. The conduction of heat from the gas to the solid, the driving force behind the flame spread, also takes place at the interfacial temperature of Tˆv, thereby making it the proper characteristic temperature at the flame leading edge. The remaining significant effect, as seen in the model sweep of Fig. 1, is due to finite-rate pyrolysis kinetics (Model 9), which has been found to affect the vaporization temperature, Tˆv, for both thick and thin fuels [4,16]. The environmental conditions including the opposing velocity, pressure, and oxygen level also influence Tˆv. However, a constant value of Tˆv, equal to 618 K, is chosen for PMMA. This value is chosen because it arises as an average value from numerical experiments covering a wide range of opposing flow velocities using the pyrolysis model of Lengelle [2,10], although the computational peak surface temperature ranges from 590 K to 700 K for the cases studied [17]. Although analytical expressions [18] can be used to calculate Tˆv, a constant
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spread rate formula, with the exception of those conditions at the higher Vˆeqv where gas-phase chemical kinetic effects come into play.
Comparison with Experimental Results and Other Theories
Fig. 4. Computations with the comprehensive model (Model 13) for two flow configurations (FP and FDC) and various environmental parameters and the prediction of the EST.
value for a given fuel appears to work reasonably well for all environmental conditions in the thermal regime. Comparison with Computation Using the Comprehensive Model (13) Computational results from the most comprehensive model (Model 13) compare well with the prediction of the final modified formula (EST Model 7) in Figs. 1 and 3. However, a mild effect of gas-phase chemistry appears in Fig. 3, where the slope of Vˆf with Vˆg for Model 13 is slightly different than the slope from the EST Model 7. To test the modified formula thoroughly, computations were carried out with the comprehensive model (Model 13) for different oxygen levels (yo,`), a wide range of opposing velocities (Vˆg), different ambient pressures (Pˆ`), different channel heights ˆ ) in the FDC configuration, and different fuel (H lengths up to the leading edge (xˆd) for the FP configuration. To compare the results from the EST and computations, we note that Eq. (4) can be rearranged as 2
FEST ˆ Vˆ f,EST 4 Veqv b5b6
(9)
The function F2EST/b5b6 carries all the thermodynamic and transport property information, and Vˆeqv carries the hydrodynamic effect making Eq. (9) a universal representation of the spread rate in the thermal regime. A plot of Vˆf,EST [b5b6/F2EST] versus Vˆeqv should be independent of the geometry (channel or flat plate) or the environmental conditions. Indeed, such is the case in Fig. 4, where the computational spread rates corresponding to the comprehensive model, Model 13, collapse around the prediction from the modified
The data of Fernandez-Pello et al. [4], obtained with PMMA in the FDC configuration, excluding the low-velocity regime influenced by buoyancy, are used here to compare different spread rate formulas. First, the data from Fig. 3 in Ref. 4 of experimental spread rate normalized with de Ris’s formula, Vˆf / Vˆf,dR,FRG, versus the Damko¨hler number of Fernandez-Pello et al., DaFRG, are presented in Fig. 5. In evaluating Vˆf,dR,FRG, we used Eq. (2) except Tˆf,ad,ST was replaced with the adiabatic, stoichiometric equilibrium temperature and the properties were evaluated at Tˆref,q 4 Tˆref,k 4 [Tf,ad,dR ` Tˆ`/2]. The data plotted in this way suggest that the thermal regime is nonexistent, and the entire data set, through its dependence on DaFRG, is kinetically controlled. When Vˆf,EST (Eq. [4]) is used to normalize the experimental spread rate, all properties evaluated at Tˆm and Vˆeqv (instead of Vˆg is used in the Damko¨hler number), the normalized data become tightly packed around unity for large DaEST, indicating the distinct presence of the thermal regime. The velocity gradient model of Wichman, obtained from a much more rigorous theory, also supports the existence of the thermal regime. However, the data at different oxygen levels do not collapse (Fig. 3 in Ref. 12, as well as in Fig. 5 in the thermal regime). It should be noted, however, that the hydrodynamic factor in the Wichman formula, the square root of the velocity gradient at the flame leading edge, scales with chyd of the EST. Spread rate in the thermal regime is independent of combustion kinetics. However, in the kinetic regime, characterized by a low DaEST, Vˆf is lowered by the finite-rate combustion kinetics. The activation temperature appears in the expression [4] for DaEST in such a way that its value affects the compactness of the data in the kinetic regime in Fig. 5. The best compactness is achieved with an activation temperature of 10,000 K and represents the best fit for the global, single-step, complete combustion reaction.
Conclusions The major drawbacks of the widely used de Ris formula for opposed-flow flame spread rate over thick fuels were examined through a numerical model study that investigated the effects of assumptions on spread rate. Details of flame structure, other than determining whether the flame hugs the surface or is lifted, were not investigated. A modified
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Fig. 5. Experimental spread rate [4], normalized by Vˆf,dR,FRG, monotonically increases with the Damkohler number in the left graph. The same data exhibit a thermal limit on the right graph when normalized with Vˆf,EST.
formula for spread rate, Eq. (9), is proposed through an EST. The significant conclusions are 1. The de Ris formula can be “in error” by as much as a factor of 10 when compared to experiments. 2. The assumptions of a slug flow, a surface-hugging flat flame, and constant properties are identified as the principal causes for this discrepancy. 3. The assumptions of infinite-rate chemistry, mass flux linearization, and neglect of radiation are found to be reasonable assumptions in the thermal regime. 4. The proposed formula, derived from an extended simplified theory, overcomes the main drawbacks of the de Ris formula and compares well with computational spread rates from a comprehensive mathematical model. 5. The formula covers two different flow configurations and can be expanded to include others. 6. It compares well with the available experimental results firmly establishing the thermal regime. 7. The formula can be used to deduce the activation energy for a global combustion reaction appropriate to flame spread modeling.
Nomenclature B cˆg cˆs DaFPRG DaEST
ˆ H K Lˆg,EST or Lˆg,ST y s T Tf
the transfer number, see Eqs. (1) and (3) (baseline value:7.24) specific heat of the gas (baseline value: 1.183 kJ/kg • K), kJ/kg • K specific heat of the solid, 1.465 kJ/ kg•K for PMMA Damko¨hler number as used in Ref. 4 Damko¨hler number as used in Ref. 4, except Vˆg is replaced with Vˆeqv, properties evaluated at Tˆv, and Tˆf,ad,ST is used for the flame temperature channel height in the FDC configuration heat transfer blockage factor, b2ln (1 ` B)/Bb4, see Eq. (2) (baseline value: 3.43) reference length scale in the gas phase, Lˆg,EST 4 aˆ g/Vˆeqv, Lˆg,ST 4 aˆ g/Vˆg species mass fraction in the gas phase stoichiometric ratio for complete combustion (baseline value: 1.92) gas temperature, Tˆ/Tˆ` flame temperature, Tˆf/Tˆ`
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Tf,ad
Tˆ` Tˆref Vˆeqv Vˆg Vˆf x, y xˆa xeig
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stoichiometric, adiabatic, flame temperature, linearized mass flux Tf,ad,ST 4 Tˆf,ad,ST/Tˆ` (de Ris formula, Eq. [2]); Tf,ad,EST 4 Tˆf,ad,EST/Tˆ`(EST, Eq. [4]) ambient temperature (baseline value: 298 K), K reference temperature for the evaluation of kˆ g (Tˆref,k) and qˆ g (Tˆref,q) equivalent opposing flow velocity, Eqs. (6) and (7), m/s velocity of the oxygen/nitrogen mixture, m/s value of the spread rate, m/s dimensionless x, y coordinates, xˆ/Lˆg and yˆ/Lˆg distance between the flame and boundary layer leading edges x location of the onset of vaporization, chosen as the origin
Greek Symbols
aˆ g bthick b1 b2 b3 b4 b5 b6 Dhˆ oc Dhˆ ov
f1 f2 f3 chyd clift kˆ g
thermal diffusivity, kˆ g/qˆ gCˆ g, m2/s dimensionless spread rate for thick fuel, Vˆf /Vˆg [qˆ sCˆs/qˆ gCˆg] stoichiometric parameter, yo,` /s (baseline value: 0.278) heat of combustion parameter, Dhˆ oc / CˆgTˆ` (baseline value: 73.468) first vaporization parameter, Tˆv /Tˆ` (baseline value: 2.07) second vaporization parameter, |Dhˆ ov|/cˆgTˆ` (baseline value: 2.669) thermal conductivity parameter, kˆ s/kˆ g (baseline value, ST: 5.595; EST:3.885) thermal capacity parameter, qˆ sCˆs / qˆ gCˆg (baseline value, ST: 1201.3; EST: 2491.3) Heat of combustion (baseline value: 25,900 kJ/kg), kJ/kg Heat of vaporization (baseline value: 941 kJ/kg), kJ/kg First conserved scalar, f1 [ 1/b4 [(T 1 1) ` b1b2(yo/yo,` 1 1)] Second conserved scalar, f2 [ 1b2/ b4 [yF 1 b1 (yoyo,` 1 1)] Third conserved scalar, f3 [ Ts 1 1/b4 Hydrodynamic coefficient used in the EST, Eqs. (6) and (7) Flame liftoff coefficient used in the EST, Eq. (8) gas-phase conductivity evaluated at Tˆr,k (baseline value: 0.0515 W/ m • K at 700 K), W/m • K
kˆ s qˆ g s
solid-phase thermal conductivity (baseline value: 0.188 W/m • K), W/m • K gas density evaluated at Tˆr,q, kg/m3 fuel half-thickness in the units of Lˆg, s [ sˆ /Lˆg
Subscripts dR FRG f g i o s v w `
de Ris Fernandez-Pello, Ray, and Glassman (Ref. 4) flame gas phase index for conserved scalars (i 4 1,2,3) and controlling parameters (i 4 1 . . . 5) oxygen solid fuel vaporization vaporizing wall ambient conditions
Superscripts ^
dimensional value Acknowledgments
This work was supported by NASA through Grant NCC3-221 and Contract NAS3-23901. We thank Sandra Olson and Kurt Sacksteder, project scientists on the grant and contract, respectively, for valuable discussions surrounding the work. We also thank Mike Delichatsios of Factory Mutual for his valuable comments regarding the flame lift coefficient.
REFERENCES 1. de Ris, J. N., Twelfth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1969, pp. 241–252. 2. West, J., Bhattacharjee, S., and Altenkirch, R. A., J. Heat Transf. 116:646–651 (1994). 3. Di Blasi, C., Combust. Flame 100:332–340 (1995). 4. Fernandez-Pello, A. C., Ray, S. R., and Glassman, I., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, pp. 579–589. 5. Wichman, I. S., Combust. Flame 50:287–304 (1983). 6. Wichman, I. S., Prog. Energy Combust. Sci. 18:553– 593 (1992). 7. Altenkirch, R. A., Eichhorn, R., and Rizvi, A. R., Combust. Sci. Technol. 32:49–66 (1983). 8. Bhattacharjee, S., West, J., and Dockter, S., Combust. Flame 104:66 – 80 (1996).
SPREAD RATE IN OPPOSED-FLOW 9. Bhattacharjee, S., Altenkirch, R. A., and Sacksteder, K., J. Heat Transf. 118:181–190 (1996). 10. Lengelle, G., AIAA J. 8:1989–1996 (1970). 11. Bhattacharjee, S., Combust. Flame 93:434–444 (1993). 12. Wichman, I. S., Williams, F. A., and Glassman, I., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, pp. 835–845. 13. de Ris, J. N., “The Spread of a Diffusion Flame over a Combustible Surface,” Ph.D. Thesis, Harvard University, 1968.
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14. Bejan, A., Convection Heat Transfer, Wiley, New York, 1982. 15. Wichman, I. S. and Williams, F. A., Combust. Sci. Technol. 32:91–123 (1983). 16. Bhattacharjee, S., Bhaskaran, K. K., and Altenkirch, R. A., Combust. Sci. Technol. 100:163–183 (1994). 17. West, J., Ph.D. Thesis, SDSU/UCSD, 1996, in preparation. 18. Fernandez-Pello, A. and Williams, F. A., Combust. Flame 28:251–277 (1977).
COMMENTS M. A. Delichatsios, FMRC, USA. This paper has a lot of strong points but it can be further improved: 1. It is not clear how to use the methodology to account for the presence of external heat flux in the pyrolysing region. 2. Reradiation losses are neglected, although they are important near extinction. 3. The determination of the lift parameter needs more clarification. It is my contention that the so called hang distance is a result of finite kinetics and is not an eigenvalue of the problem for infinitely fast kinetics [1,2].
REFERENCES 1. Delichatsios, M. A., Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1495–1503. 2. Chen, Y. and Delichatsios, M. A., Combust. Flame, 99:601–609 (1994). Author’s Reply. First, this methodology was not intended to provide for an explicit means for accounting for an external heat flux, and so the presence of an external heat flux is not included. Second, the neglect of reradiation is justified in Fig. 1 where the addition of reradiation in the numerical model from Model 9 to Model 10 does not cause a large change in the flame spread rate. There is evidence [1] that reradiation losses provide a finite impact on the flame spread rate even far away from extinction. However, this effect is small compared to other effects such as the boundary layer character and variable gas-phase properties as discussed in the text of the paper. The Extended Simplified Theory is intended for use under conditions far away from extinction, i.e., for use in the
Thermal Regime. In Fig. 3 of the paper the prediction of the Extended Simplified Theory compares reasonably well with the computed results of Model 13, which includes radiation in the gas phase as well as reradiation. Consequently, the EST performs quite well with the neglect of reradiation and its inclusion would not improve the accuracy of the theory. Finally, the determination of the lift parameter, clift, is given by Eq. (8) to within an unknown constant. By forcing the flame locations predicted by the theory to coincide with the flame locations computed from the numerical model the value of the unknown constant can be determined. The “hang distance” has been neglected in this work. The flame is assumed to touch the fuel surface at the location of the onset of pyrolysis, the coordinate origin. The results of numerical computation using Model 4 for thermally thick fuels in which the hang distance was not present is a justification for the neglect. The existence of the hang distance in numerical models using infinite rate chemical kinetics with thermally thin fuels has already been demonstrated [2]. While the numerical results here for Model 4 do not produce a hang distance for thermally thick fuels, we have found evidence numerically that a hang distance may be possible even when using infinite rate chemical kinetics, but for models more complex than Model 4. There is an ongoing search to learn more about the hang distance, but this search does not affect the current simplified theory because the hang distance is neglected in this theory.
REFERENCES 1. West, J., Bhattacharjee, S., and Altenkirch, R. A., J. Heat Trans. 116:646–651 (1994). 2. Bhattacharjee, S., Combust. Flame 93:434–444 (1993).
DETERMINATION OF THE SPREAD RATE IN OPPOSED-FLOW FLAME SPREAD OVER THICK SOLID FUELS IN THE THERMAL REGIME SUBRATA BHATTACHARJEE and JEFF WEST Department of Mechanical Engineering San Diego State University San Diego, CA 92182, USA ROBERT A. ALTENKIRCH School of Mechanical and Materials Engineering Washington State University Pullman, WA 99164, USA
Limitations of the assumptions in the classic de Ris formula [1] for opposed-flow flame spread rate over thick fuels in the thermal regime are explored through numerical experiments in which the assumptions used in the original theory are eliminated one by one from the mathematical model. While the assumptions of infinite-rate chemistry, mass flux linearization, and neglect of radiation are found to be reasonable in the thermal regime, assumptions of a slug flow, a surface-hugging flame, and constant properties are responsible for the predictions of spread rate being in error by as much as a factor of 10. A spread rate formula, derived from a semiempirical solution of a simplified problem, is shown to overcome the drawbacks of the de Ris formula while retaining its simplicity. A hydrodynamic coefficient, developed for two different flow configurations, and a flame liftoff coefficient, both obtainable from the known parameters of the problem, correct for some of the major limiting assumptions in the de Ris theory. The formula developed here, which can be written as 2
FEST ˆ Vˆ f,EST 4 Veqv b5b6 in which Vˆeqv contains the hydrodynamic information and the factor in front of it the chemical information, performs significantly better than the de Ris formula when tested against available data and extensive computational results for different flow configurations, opposed-flow velocities, oxygen levels, and ambient pressures.
Introduction Flame spread over a thermally thick fuel can be classified into three distinct regimes based on the strength of the opposing flow (1) the low-velocity quenching regime in which radiative effects are important [2,3], (2) the high-velocity blow-off regime in which chemical kinetic effects are important [4], and (3) the moderate-velocity thermal regime [1,5] in which both the radiative and chemical kinetic effects are relatively unimportant. Of these, only the thermal regime, because of its simplified physics, has yielded useful, closed-form solutions to model problems for the spread rate. The classic de Ris formula for the spread rate in the thermal regime, based on a number of symplifying assumptions, is probably the most well-known formula in the opposed-flow flame spread literature [6]. It has found use beyond characterization [4] of the thermal regime in which by comparison of the
spread rate with the de Ris formula attempts have been made [4,7] to identify the particular physics that delineate new regimes. The Damko¨hler number correlations of Fernandez-Pello et al. [4] and Altenkirch et al. [7] are some cases in point. To date, only a handful of studies have attempted to assess the impact of the various assumptions on the spread rate formula, the most notable being the work of Wichman [5], who showed that removal of the Oseen approximation can cause a significant difference in the flame spread rate. The Wichman formula, however, contains two undetermined constants and a velocity gradient that is not readily known. Here, we undertake a systematic evaluation through numerical case studies of the effects of the entire set of assumptions used by de Ris. We identify three major drawbacks of the de Ris theory that can cause large errors (a few hundred percent) in the prediction of spread rate. A simplified theory is
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Fig. 1. Computed and predicted (EST) spread rates for the baseline case (50/50 O2/N2 by volume, PMMA, 40 cm/ s, FDC) corresponding to different models with the physics added for each model change as follows: Model 1, de Ris model (zero hang distance); Model 2, nonzero hang distance; Model 3, Vˆrelative 4 Vˆg ` Vˆf ; Model 4, non-Oseen flow (i.e., x momentum equation solved); Model 5, wall blowing (i.e., y momentum equation solved); Model 6, variable gas density; Model 7, variable transport properties; Model 8, finite-rate combustion kinetics; Model 9, finiterate pyrolysis kinetics; Models 10–13, models with reradiation, gas radiation, and radiation feedback; Model 14, experiment from Ref. 4.
developed, and a semi-empirical, modified spread rate expression is proposed that retains the elegant simplicity of the de Ris formula yet corrects for its severe assumptions. Two flow configurations are investigated: (1) Fully developed channel (FDC) flow with the channel height much larger than the flame standoff distance, that is, the centerline velocity is assumed to remain unaffected by the spreading flame, and (2) forced flow over a flat plate (FP). Mathematical Models The mathematical models are listed in Fig. 1, with Model 1 corresponding to the classic de Ris formula and Model 14 representing the experimental data of Fernandez-Pello et al. [4] obtained in the FDC configuration. The spread rates in Fig. 1 are for a baseline case, that is, spread over a thick slab of PMMA flush mounted on the wall of a two-dimensional channel with an opposing, fully developed flow of a 50/50 O2/N2 mixture, by volume, with a centerline velocity of 40 cm/s at 1 atm pressure. These conditions are chosen to duplicate experimental conditions for comparison and to ensure that the case belongs to the thermal regime. In the numerical solution [8] (Model 2) to the de Ris problem [1], the flame front is not assumed to coincide with the pyrolysis front. The major assump-
tions are gradually removed in the subsequent models: (1) in Models 3, 4, and 5, hydrodynamic effects are included, (2) in Models 6 and 7, hot-gas expansion and variable properties are introduced, (3) in Models 8 and 9, finite-rate gas-phase and solid-phase chemistry are introduced, and (4) in Models 10 through 13, surface and gas-phase radiation are introduced. Steady-state equations solved in the most comprehensive model (Model 13) include the energy equation in the solid phase, and continuity, x- and y- momentum, species equations for fuel, oxygen, and nitrogen, and energy in the gas phase, the ideal gas equation and a square root dependence of transport properties on temperature. Major assumptions that remain in Model 13 include unit Lewis number, constant molecular weight and specific heats, single-step chemistry for combustion and pyrolysis, constant solid density, no surface regression, and no soot production. The submodels used in this model [9] comprise Lengelle’s pyrolysis model [2,10], Arrhenius second-order kinetics for the gas phase, and the radiation model of Bhattacharjee and Altenkirch [9]. An iterative scheme [9] in which the gas- and the solid-phase equations are alternately solved using the SIMPLER algorithm is used for the numerical solution of the spread rate and the conserved variables. Details of the mathematical formulation, numerical solution procedure, and code validation can be found elsewhere [9,11] and are omitted here for brevity. On the very left of the abscissa in Fig. 1 is the exact solution of de Ris, and on the extreme right is the experimental result of Fernandez-Pello et al. [4], with the intermediate points being the computational solutions with the assumptions of de Ris removed one by one as listed in Fig. 1. In evaluating the spread rate with the de Ris formula, the properties are evaluated at the ambient temperature, and the linearized, adiabatic temperature is used for the flame temperature. The hang distance, the distance between the flame leading edge and the pyrolysis front, is postulated to be zero in the de Ris formulation (Model 1) [11], although the underlying mathematical model allows for the fuel to diffuse upstream to form a finite hang distance. For a thin fuel, this inconsistency in the “exact” solution [11] was found to have a significant effect on Vˆf. However, for the thick fuel, with the flame computed to be on the surface as de Ris had postulated, the hang distance is found to be zero in the numerical solution, and the exact solution is reproduced (within 0.01%) in Model 2. In Model 3, the velocity the flame encounters is modified from Vˆg, used in the de Ris model (Model 1), to Vˆrel 4 Vˆg ` Vˆf, in flame-fixed coordinates. The spread rate does not change significantly because Vˆg k Vˆf in the thermal regime. A large change in Vˆf occurs when the momentum equations are
SPREAD RATE IN OPPOSED-FLOW
solved (Model 4) because the effective opposing velocity near the flame leading edge is reduced drastically by the formation of the boundary layer. Removal of mass transfer linearization by including surface blowing (Model 5) does not change Vˆf appreciably. Variable gas density and thermal conductivity (Models 6 and 7) have significant effects. In these models, the ideal gas equation is used to compute local density while a square root dependence on temperature is used to evaluate the thermal conductivity and viscosity. Finite-rate combustion chemistry (Model 8), for the kinetic constants used here, has no effect in the thermal regime. However, pyrolysis kinetics (Model 9) have a significant effect on Tˆv and hence the spread rate. Different radiation models [9], surface reradiation (Model 10), gas radiation loss (Model 11), gas and surface radiation loss (Model 12), and gas and surface radiation with radiation feedback from the gas to the solid (Model 13) affect the flame size and structure significantly. However, the spread rate appears to remain largely unaffected. Numerical results also establish that the dependence of Vˆf on the opposing flow velocity, Vˆg, deviates significantly from the linear relationship of the de Ris formula. With the momentum equations included (Model 4), the dependence of Vˆf on Vˆg for the FP and the FDC flow configurations are quite different, which is missed by the de Ris formula. The Wichman formula accounts for this geometrical dependence through a velocity gradient term, which is difficult to evaluate without experimentally determined velocity profiles. A major drawback of the de Ris theory stems from its prediction of a surface-hugging flame [8,12] that releases just enough heat to sustain a flame temperature equal to the vaporization temperature. Once the Oseen approximation is removed (Model 4), the flame lifts from the surface, and the flame temperature rapidly increases in the downstream direction. Flame liftoff is accompanied by a reduction in the spread rate, which is a different effect from the hydrodynamic effects discussed later. Finally, the existing spread rate formulas suffer from inconsistent use of three characteristic temperatures, the linearized adiabatic flame temperature, Tˆf,ad,lin, and the reference temperatures for evaluation of gas density and thermal conductivity, Tˆref,q and Tˆref,k, respectively. Use of these temperatures varies widely, resulting in disparate prediction of spread rate for the same case. For example, for the same baseline case of Fig. 1, the spread rates from the de Ris formula from different sources are: 14.5 mm/s (Eq. [6]), 6.4 mm/s [12], and 1.59 mm/s [4]. The constant property results of Fig. 1 were obtained with Tˆref,q 4 Tˆref,k 4 Tˆ`. Any modified formula must clearly specify these temperatures.
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Simplified and Extended Simplified Theory (ST and EST) As a guide in developing the EST spread rate formula, it is instructive to revisit the de Ris model and a simplified solution [11] for the spread rate (ST). The governing equations for the Schvab–Zeldovich conserved scalars, the solid-phase temperature, and the energy balance at the vaporizing interface, with mass-transfer linearization as used by de Ris [1], can be written in terms of six known parameters, b1 . . . b6, which are, in order, yo,` , s
Dhˆ oc Tˆ v , , cˆgTˆ ` Tˆ `
|Dhˆ ov| kˆ s qˆ sCˆ s , , cˆgTˆ ` kˆ g qˆ gCˆ g
To obtain an approximate solution in the downstream region, the boundary conditions are simplified by assuming that the flame lies along the vaporizing surface so that yF 4 yo 4 0 there. In addition, the governing differential equations are made parabolic, resulting in the following solutions for the coupling functions in the downstream region from the flame leading edge:
f1,ST b 4 4 f2,ST 4 erfc(gST) 1B b1b2 f3,ST 4
b3 1 1 erfc 1gST b4
1
where gST [
!
2
bthick,ST b5
yˆ ˆ 2!Lg,ST(xˆeig 1 xˆ)
(1)
Upon substitution of these expressions into the interface energy balance, an expression for the dimensionless spread rate is obtained, which is identical to the de Ris formula: bthick,ST 4
F2ST b5
where FST [
Tˆ f,ad,ST 1 Tˆ v Tf,ad,ST 1 b3 ˆTv 1 Tˆ ` 4 b3 1 1
Tf,ad,ST 4 1 ` b1(b2 1 Kb4)
(2)
Here, Tˆf,ad,ST is the asymptotic, adiabatic, stoichiometric flame temperature with mass flux linearization that appears in the de Ris formula. The simplified solution to the de Ris model, although valid only in the downstream region, provides insight into the limitations of the model and, when compared with numerical solution of more advanced models, leads to the corrections developed here. Lifted Flame and Hydrodynamic Corrections From the computed solution of the baseline case for four different models (Models 2–5) of Fig. 1, the
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here, Vˆg is replaced with an equivalent velocity, Vˆeqv 4 chydVˆg, where chyd, a second unknown coefficient at this point, is to contain the information about the character of the opposing flow at the flame leading edge. With introduction of clift and chyd, the problem admits a solution, the EST, of the same form as the ST with Lˆg,EST [ aˆ g/chydVˆg f1,EST b4 4 f2,EST 4 erfc(gEST) 1B cliftb1b2 f3,EST 4
Fig. 2. Computed surface profile of the second coupling function for different models for the baseline case. Models 4 and 5 approach their asymptotic values in the downstream direction.
interfacial value of the coupling function f2, normalized by its value at the flame, f2,flame 4 b1 b2/b4 4 7.65, is plotted in Fig. 2. As expected for the de Ris model (Model 2), f2/f2,flame assumes a constant value for x , 0 as the flame lies along the vaporizing surface. The correction of the velocity boundary condition (Model 3) does not seem to have any effect on this profile. However, as the slug flow approximation is replaced with a fully developed velocity profile in Model 4, and then the mass transfer linearization at the interface is replaced with the proper blowing term in Model 5, the f2 profiles change dramatically. Ahead of the flame (x . 0 in Fig. 2), f2 decreases more gently, as the distance over which species diffuse ('aˆ g/Vˆg) increases because of a lower opposing velocity near the wall. Downstream f2 increases sharply near the leading edge and continues to increase gently toward the corresponding asymptotic value [13] of ln(1 ` B) f2,flame (4 7.6f2,flame) b1 for Model 4 and B(1 ` b1) f (4 4.0f2,flame) b1(1 ` B) 2,flame for Model 5. The flame is lifted in both Model 4 and Model 5. To account for this in the EST, we write f2,wall,EST 4 cliftf2,flame, where the coefficient clift, an unknown at this point, is to be correlated with known parameters. The Oseen flow approximation (i.e., the assumption of a slug flow) is responsible for the proportionality between Vˆf and Vˆg, which is seen to be absorbed in the definition of bthick in the de Ris formula (bthick,ST 4 F2ST/b5). In the modified formulation
where gEST [
b3 1 1 b4
1
erfc 1gEST
!
2
bthick,EST b5
yˆ
(3)
2!Lˆ g,EST(xˆeig 1 xˆ)
bthick,EST 4 chyd
F2EST b5
where FEST [
Tf,ad,EST 1 b3 b3 1 1
Tf,ad,EST 4 1 ` b1(b2 1 cliftKb4) (4) The preceding reduces to the de Ris solution for chyd 4 clift 4 1. However, a number of favorable consequences results from a clift . 1 and chyd , 1. First, the spread rate is reduced compared to the de Ris prediction, as can be verified by comparing Eqs. (4) and (2). Second, the length scale is enlarged (Lˆg,EST 4 Lˆg,ST/chyd), which is consistent with the enlargement of the preheat zone for Model 4 in Fig. 2. Third, the flame is lifted, and the flame location can be found from Eq. (3) to be the locus of erfc(gEST,flame) 4 1/clift. The flame temperature can be obtained from the solution for the first coupling function f1,EST:
1
2
1 (5) clift For clift 4 1, Tf,EST reduces to the vaporization temperature b3 consistent with ST and the de Ris model. Note that it is the adiabatic flame temperature, Tˆf,ad,EST, and not the actual flame temperature, that appears in the spread rate formula, Eq. (4). Tf,EST 4
b3
clift
` (1 ` b1b2) 1 1
Evaluation of the Coefficients chyd and clift To develop a functional relation for chyd, attention is focused on the velocity profile encountered by the flame leading edge submerged in a boundary layer. Because the length scale there is Lˆg,EST, the proper scale for Vˆeqv is postulated to be proportional to the velocity at a distance Lˆg,EST from the fuel surface at the leading edge [14]. Depending on the flow configuration of interest, Vˆeqv and, hence, chyd, can be written as
SPREAD RATE IN OPPOSED-FLOW
Fig. 3. Computations and the EST prediction for various mathematical models for the FDC configuration.
aˆ g 1.5Vˆ g ˆ FDC: Vˆ eqv ; ˆ Lg,EST ⇒ chyd,FDC ; ˆ ˆ (H/2) HVg
1/2
1 2
aˆ 4 ch,FDC ˆ ˆg HVg
1/2
1 2
1/2 Pr11/2 4 ch,FDC Re1 H
(6)
aˆ g Vˆ g ˆ FP: Vˆ eqv ; Lg,EST ⇒ chyd,FP ; dˆ xˆdVˆ gPr
1
4 ch,FP
aˆ g
1/4
1xˆ Vˆ Pr2 d g
1/4
2
1/4 Pr11/2 4 ch,FP Re1 xd
(7) Although the hydrodynamic constants, ch,FDC and ch,FP, remain to be determined, the usefulness of these expressions is evident when substituted into the spread rate formula, Eq. (4). The dependence of Vˆf on Vˆg is correctly predicted by this modified equation. To obtain a functional relationship between clift and the governing parameters of the EST, note that in the far downstream region the asymptotic solution corresponding to Model 4 yields a value of ln(1 ` B)/b1 for clift. Therefore, clift can be expected to depend on this parameter. A selected set of computations, covering a wide range of the thermal regime, both for the channel flow and the flat plate configurations, were carried out using Model 4 (i.e., the model that includes the momentum equations without wall blowing). By fitting the flame location predicted by the EST, clift can be calculated for the baseline case. The entire data set indicates a linear relationship independent of the flow configuration and environmental conditions. A least-square-error fit yields: clift 4 1 ` clift
B(1 ` b1) , clift 4 0.09 b1(1 ` B)
(8)
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By matching the computed and predicted spread rates, ch,FDC and ch,FP are obtained and are found to be almost constants as expected. The best-fit values are ch,FDC 4 7.65 for the channel flow and ch,FP 4 2.65 for the flat plate geometry. The flat plate value is understandably smaller because the boundary layer thickness for this geometry is significantly underestimated by scaling whereas for the fully developed channel flow it is known exactly. Computations of flame shapes show that blowing does not appear to have much impact, especially in the FDC configuration; the same is true for the spread rate as shown in Fig. 1. The EST developed for Model 4, then, is expected to work for Model 5. To incorporate the variable density effect (Model 6), which results in a significant decrease in Vˆf without much effect on the slope of Vˆf versus Vˆg as can be seen in Fig. 3, the gas density in the EST is evaluated at Tˆref,q 4 Tˆv. Good agreement is obtained between the prediction and Model 6 computations for the baseline case of Fig. 1 and the test cases of Fig. 3. When thermal conductivity is allowed to vary with the square root of the local temperature (Model 7), Vˆf increases sharply as shown in Figs. 1 and 3. This increase is found to be independent of the ambient oxygen level. Therefore, once again, a reference temperature Tˆref,k 4 Tˆv is chosen for the evaluation of thermal conductivity. The resulting modified formula (EST Model 7), although an improvement, still underpredicts (Fig. 1) the spread rate. It is not surprising that the vaporization temperature, Tˆv, turns out to be a reasonable temperature for the evaluation of properties, as opposed to the adiabatic, stoichiometric flame temperatures based on complete or equilibrium combustion [4,5,15]. Near the flame front, the flame temperature rapidly decreases as the leading edge is approached, and it is almost equal to Tˆv at the leading edge. The conduction of heat from the gas to the solid, the driving force behind the flame spread, also takes place at the interfacial temperature of Tˆv, thereby making it the proper characteristic temperature at the flame leading edge. The remaining significant effect, as seen in the model sweep of Fig. 1, is due to finite-rate pyrolysis kinetics (Model 9), which has been found to affect the vaporization temperature, Tˆv, for both thick and thin fuels [4,16]. The environmental conditions including the opposing velocity, pressure, and oxygen level also influence Tˆv. However, a constant value of Tˆv, equal to 618 K, is chosen for PMMA. This value is chosen because it arises as an average value from numerical experiments covering a wide range of opposing flow velocities using the pyrolysis model of Lengelle [2,10], although the computational peak surface temperature ranges from 590 K to 700 K for the cases studied [17]. Although analytical expressions [18] can be used to calculate Tˆv, a constant
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spread rate formula, with the exception of those conditions at the higher Vˆeqv where gas-phase chemical kinetic effects come into play.
Comparison with Experimental Results and Other Theories
Fig. 4. Computations with the comprehensive model (Model 13) for two flow configurations (FP and FDC) and various environmental parameters and the prediction of the EST.
value for a given fuel appears to work reasonably well for all environmental conditions in the thermal regime. Comparison with Computation Using the Comprehensive Model (13) Computational results from the most comprehensive model (Model 13) compare well with the prediction of the final modified formula (EST Model 7) in Figs. 1 and 3. However, a mild effect of gas-phase chemistry appears in Fig. 3, where the slope of Vˆf with Vˆg for Model 13 is slightly different than the slope from the EST Model 7. To test the modified formula thoroughly, computations were carried out with the comprehensive model (Model 13) for different oxygen levels (yo,`), a wide range of opposing velocities (Vˆg), different ambient pressures (Pˆ`), different channel heights ˆ ) in the FDC configuration, and different fuel (H lengths up to the leading edge (xˆd) for the FP configuration. To compare the results from the EST and computations, we note that Eq. (4) can be rearranged as 2
FEST ˆ Vˆ f,EST 4 Veqv b5b6
(9)
The function F2EST/b5b6 carries all the thermodynamic and transport property information, and Vˆeqv carries the hydrodynamic effect making Eq. (9) a universal representation of the spread rate in the thermal regime. A plot of Vˆf,EST [b5b6/F2EST] versus Vˆeqv should be independent of the geometry (channel or flat plate) or the environmental conditions. Indeed, such is the case in Fig. 4, where the computational spread rates corresponding to the comprehensive model, Model 13, collapse around the prediction from the modified
The data of Fernandez-Pello et al. [4], obtained with PMMA in the FDC configuration, excluding the low-velocity regime influenced by buoyancy, are used here to compare different spread rate formulas. First, the data from Fig. 3 in Ref. 4 of experimental spread rate normalized with de Ris’s formula, Vˆf / Vˆf,dR,FRG, versus the Damko¨hler number of Fernandez-Pello et al., DaFRG, are presented in Fig. 5. In evaluating Vˆf,dR,FRG, we used Eq. (2) except Tˆf,ad,ST was replaced with the adiabatic, stoichiometric equilibrium temperature and the properties were evaluated at Tˆref,q 4 Tˆref,k 4 [Tf,ad,dR ` Tˆ`/2]. The data plotted in this way suggest that the thermal regime is nonexistent, and the entire data set, through its dependence on DaFRG, is kinetically controlled. When Vˆf,EST (Eq. [4]) is used to normalize the experimental spread rate, all properties evaluated at Tˆm and Vˆeqv (instead of Vˆg is used in the Damko¨hler number), the normalized data become tightly packed around unity for large DaEST, indicating the distinct presence of the thermal regime. The velocity gradient model of Wichman, obtained from a much more rigorous theory, also supports the existence of the thermal regime. However, the data at different oxygen levels do not collapse (Fig. 3 in Ref. 12, as well as in Fig. 5 in the thermal regime). It should be noted, however, that the hydrodynamic factor in the Wichman formula, the square root of the velocity gradient at the flame leading edge, scales with chyd of the EST. Spread rate in the thermal regime is independent of combustion kinetics. However, in the kinetic regime, characterized by a low DaEST, Vˆf is lowered by the finite-rate combustion kinetics. The activation temperature appears in the expression [4] for DaEST in such a way that its value affects the compactness of the data in the kinetic regime in Fig. 5. The best compactness is achieved with an activation temperature of 10,000 K and represents the best fit for the global, single-step, complete combustion reaction.
Conclusions The major drawbacks of the widely used de Ris formula for opposed-flow flame spread rate over thick fuels were examined through a numerical model study that investigated the effects of assumptions on spread rate. Details of flame structure, other than determining whether the flame hugs the surface or is lifted, were not investigated. A modified
SPREAD RATE IN OPPOSED-FLOW
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Fig. 5. Experimental spread rate [4], normalized by Vˆf,dR,FRG, monotonically increases with the Damkohler number in the left graph. The same data exhibit a thermal limit on the right graph when normalized with Vˆf,EST.
formula for spread rate, Eq. (9), is proposed through an EST. The significant conclusions are 1. The de Ris formula can be “in error” by as much as a factor of 10 when compared to experiments. 2. The assumptions of a slug flow, a surface-hugging flat flame, and constant properties are identified as the principal causes for this discrepancy. 3. The assumptions of infinite-rate chemistry, mass flux linearization, and neglect of radiation are found to be reasonable assumptions in the thermal regime. 4. The proposed formula, derived from an extended simplified theory, overcomes the main drawbacks of the de Ris formula and compares well with computational spread rates from a comprehensive mathematical model. 5. The formula covers two different flow configurations and can be expanded to include others. 6. It compares well with the available experimental results firmly establishing the thermal regime. 7. The formula can be used to deduce the activation energy for a global combustion reaction appropriate to flame spread modeling.
Nomenclature B cˆg cˆs DaFPRG DaEST
ˆ H K Lˆg,EST or Lˆg,ST y s T Tf
the transfer number, see Eqs. (1) and (3) (baseline value:7.24) specific heat of the gas (baseline value: 1.183 kJ/kg • K), kJ/kg • K specific heat of the solid, 1.465 kJ/ kg•K for PMMA Damko¨hler number as used in Ref. 4 Damko¨hler number as used in Ref. 4, except Vˆg is replaced with Vˆeqv, properties evaluated at Tˆv, and Tˆf,ad,ST is used for the flame temperature channel height in the FDC configuration heat transfer blockage factor, b2ln (1 ` B)/Bb4, see Eq. (2) (baseline value: 3.43) reference length scale in the gas phase, Lˆg,EST 4 aˆ g/Vˆeqv, Lˆg,ST 4 aˆ g/Vˆg species mass fraction in the gas phase stoichiometric ratio for complete combustion (baseline value: 1.92) gas temperature, Tˆ/Tˆ` flame temperature, Tˆf/Tˆ`
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Tf,ad
Tˆ` Tˆref Vˆeqv Vˆg Vˆf x, y xˆa xeig
FIRE SAFETY
stoichiometric, adiabatic, flame temperature, linearized mass flux Tf,ad,ST 4 Tˆf,ad,ST/Tˆ` (de Ris formula, Eq. [2]); Tf,ad,EST 4 Tˆf,ad,EST/Tˆ`(EST, Eq. [4]) ambient temperature (baseline value: 298 K), K reference temperature for the evaluation of kˆ g (Tˆref,k) and qˆ g (Tˆref,q) equivalent opposing flow velocity, Eqs. (6) and (7), m/s velocity of the oxygen/nitrogen mixture, m/s value of the spread rate, m/s dimensionless x, y coordinates, xˆ/Lˆg and yˆ/Lˆg distance between the flame and boundary layer leading edges x location of the onset of vaporization, chosen as the origin
Greek Symbols
aˆ g bthick b1 b2 b3 b4 b5 b6 Dhˆ oc Dhˆ ov
f1 f2 f3 chyd clift kˆ g
thermal diffusivity, kˆ g/qˆ gCˆ g, m2/s dimensionless spread rate for thick fuel, Vˆf /Vˆg [qˆ sCˆs/qˆ gCˆg] stoichiometric parameter, yo,` /s (baseline value: 0.278) heat of combustion parameter, Dhˆ oc / CˆgTˆ` (baseline value: 73.468) first vaporization parameter, Tˆv /Tˆ` (baseline value: 2.07) second vaporization parameter, |Dhˆ ov|/cˆgTˆ` (baseline value: 2.669) thermal conductivity parameter, kˆ s/kˆ g (baseline value, ST: 5.595; EST:3.885) thermal capacity parameter, qˆ sCˆs / qˆ gCˆg (baseline value, ST: 1201.3; EST: 2491.3) Heat of combustion (baseline value: 25,900 kJ/kg), kJ/kg Heat of vaporization (baseline value: 941 kJ/kg), kJ/kg First conserved scalar, f1 [ 1/b4 [(T 1 1) ` b1b2(yo/yo,` 1 1)] Second conserved scalar, f2 [ 1b2/ b4 [yF 1 b1 (yoyo,` 1 1)] Third conserved scalar, f3 [ Ts 1 1/b4 Hydrodynamic coefficient used in the EST, Eqs. (6) and (7) Flame liftoff coefficient used in the EST, Eq. (8) gas-phase conductivity evaluated at Tˆr,k (baseline value: 0.0515 W/ m • K at 700 K), W/m • K
kˆ s qˆ g s
solid-phase thermal conductivity (baseline value: 0.188 W/m • K), W/m • K gas density evaluated at Tˆr,q, kg/m3 fuel half-thickness in the units of Lˆg, s [ sˆ /Lˆg
Subscripts dR FRG f g i o s v w `
de Ris Fernandez-Pello, Ray, and Glassman (Ref. 4) flame gas phase index for conserved scalars (i 4 1,2,3) and controlling parameters (i 4 1 . . . 5) oxygen solid fuel vaporization vaporizing wall ambient conditions
Superscripts ^
dimensional value Acknowledgments
This work was supported by NASA through Grant NCC3-221 and Contract NAS3-23901. We thank Sandra Olson and Kurt Sacksteder, project scientists on the grant and contract, respectively, for valuable discussions surrounding the work. We also thank Mike Delichatsios of Factory Mutual for his valuable comments regarding the flame lift coefficient.
REFERENCES 1. de Ris, J. N., Twelfth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1969, pp. 241–252. 2. West, J., Bhattacharjee, S., and Altenkirch, R. A., J. Heat Transf. 116:646–651 (1994). 3. Di Blasi, C., Combust. Flame 100:332–340 (1995). 4. Fernandez-Pello, A. C., Ray, S. R., and Glassman, I., Eighteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1981, pp. 579–589. 5. Wichman, I. S., Combust. Flame 50:287–304 (1983). 6. Wichman, I. S., Prog. Energy Combust. Sci. 18:553– 593 (1992). 7. Altenkirch, R. A., Eichhorn, R., and Rizvi, A. R., Combust. Sci. Technol. 32:49–66 (1983). 8. Bhattacharjee, S., West, J., and Dockter, S., Combust. Flame 104:66 – 80 (1996).
SPREAD RATE IN OPPOSED-FLOW 9. Bhattacharjee, S., Altenkirch, R. A., and Sacksteder, K., J. Heat Transf. 118:181–190 (1996). 10. Lengelle, G., AIAA J. 8:1989–1996 (1970). 11. Bhattacharjee, S., Combust. Flame 93:434–444 (1993). 12. Wichman, I. S., Williams, F. A., and Glassman, I., Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1982, pp. 835–845. 13. de Ris, J. N., “The Spread of a Diffusion Flame over a Combustible Surface,” Ph.D. Thesis, Harvard University, 1968.
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14. Bejan, A., Convection Heat Transfer, Wiley, New York, 1982. 15. Wichman, I. S. and Williams, F. A., Combust. Sci. Technol. 32:91–123 (1983). 16. Bhattacharjee, S., Bhaskaran, K. K., and Altenkirch, R. A., Combust. Sci. Technol. 100:163–183 (1994). 17. West, J., Ph.D. Thesis, SDSU/UCSD, 1996, in preparation. 18. Fernandez-Pello, A. and Williams, F. A., Combust. Flame 28:251–277 (1977).
COMMENTS M. A. Delichatsios, FMRC, USA. This paper has a lot of strong points but it can be further improved: 1. It is not clear how to use the methodology to account for the presence of external heat flux in the pyrolysing region. 2. Reradiation losses are neglected, although they are important near extinction. 3. The determination of the lift parameter needs more clarification. It is my contention that the so called hang distance is a result of finite kinetics and is not an eigenvalue of the problem for infinitely fast kinetics [1,2].
REFERENCES 1. Delichatsios, M. A., Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1495–1503. 2. Chen, Y. and Delichatsios, M. A., Combust. Flame, 99:601–609 (1994). Author’s Reply. First, this methodology was not intended to provide for an explicit means for accounting for an external heat flux, and so the presence of an external heat flux is not included. Second, the neglect of reradiation is justified in Fig. 1 where the addition of reradiation in the numerical model from Model 9 to Model 10 does not cause a large change in the flame spread rate. There is evidence [1] that reradiation losses provide a finite impact on the flame spread rate even far away from extinction. However, this effect is small compared to other effects such as the boundary layer character and variable gas-phase properties as discussed in the text of the paper. The Extended Simplified Theory is intended for use under conditions far away from extinction, i.e., for use in the
Thermal Regime. In Fig. 3 of the paper the prediction of the Extended Simplified Theory compares reasonably well with the computed results of Model 13, which includes radiation in the gas phase as well as reradiation. Consequently, the EST performs quite well with the neglect of reradiation and its inclusion would not improve the accuracy of the theory. Finally, the determination of the lift parameter, clift, is given by Eq. (8) to within an unknown constant. By forcing the flame locations predicted by the theory to coincide with the flame locations computed from the numerical model the value of the unknown constant can be determined. The “hang distance” has been neglected in this work. The flame is assumed to touch the fuel surface at the location of the onset of pyrolysis, the coordinate origin. The results of numerical computation using Model 4 for thermally thick fuels in which the hang distance was not present is a justification for the neglect. The existence of the hang distance in numerical models using infinite rate chemical kinetics with thermally thin fuels has already been demonstrated [2]. While the numerical results here for Model 4 do not produce a hang distance for thermally thick fuels, we have found evidence numerically that a hang distance may be possible even when using infinite rate chemical kinetics, but for models more complex than Model 4. There is an ongoing search to learn more about the hang distance, but this search does not affect the current simplified theory because the hang distance is neglected in this theory.
REFERENCES 1. West, J., Bhattacharjee, S., and Altenkirch, R. A., J. Heat Trans. 116:646–651 (1994). 2. Bhattacharjee, S., Combust. Flame 93:434–444 (1993).