A Semi-Analytical Variable Property Droplet Combustion Model · dr = _m i (1a) X h im_ i 24ˇrk dT...

Paper # 070HE-0396 Topic: Hydrocarbon Droplet Combustion

8th US National Combustion MeetingOrganized by the Western States Section of the Combustion Institute

and hosted by the University of UtahMay 19-22, 2013.

A Semi-Analytical Variable Property Droplet Combustion Model

John Sisti1 Paul E. DesJardin1

1Department of Mechanical and Aerospace Engineering,University at Buffalo, The State University of New York

Buffalo, NY 14260-4400

A multi-zone droplet burn model is developed to account for changes in the thermal and transportproperties as a function of droplet radius. The formulation is semi-analytical - allowing for accurateand computationally efficient estimates of flame structure and burn-rates. Zonal thermal and transportproperties are computed using the Cantera software and pre-tabulated for rapid evaluation during run-time. Model predictions are compared to experimental measurements of burning n-heptane, ethanoland methanol droplets. An adaptive zone refinement algorithm is developed that minimizes the numberof zones required to provide accurate estimates of burn time without excess zones. A sensitivity studyof burn-rate and flame stand-off with far-field oxygen concentration is conducted with comparisons toexperimental data. Overall agreement to data is encouraging with errors typically less than 20% forpredictions of burn-rates, stand-off ratio and flame temperature for the fuels considered.

1 Introduction

The topic of droplet combustion has a rich history. Reviews and texts on droplet combustion in-clude those of Faeth [1, 2], Sirignano [3, 4], Williams [5, 6], Law [7], Kuo [8], Turns [9], Glassman[10]. Early theories of droplet burning came from Spaulding [11], Godsave [12] and Lorell et al.[13, 14] and were based on either single or two-zone descriptions with constant properties withineach zone. While these early theories are elegant and relatively simple to formulate, they are alsowell known to greatly over-predict the flame radius and under-predict burn times [7, 15]. Sinceflame radii and burn rates are a major source of data from droplet experiments, it is important tohave a droplet burn model that is sufficiently accurate to predict these quantities. The major sourceof the modeling assumption error leading to the exaggerated flame radius is the dependence of ther-mal and transport properties with radius, stemming from changes in composition and temperature.Many enhancements to these early theories have improved the description of property variationeither through improved analytics [16, 17] or direct numerical integration [18].

For the current effort, it is highly desirable the droplet model be computationally efficient sinceits intended use is in a subgrid scale (SGS) model for large eddy simulations of turbulent reacting,multiphase flows. Current models for droplet burning usually fall into two categories. The firstare detailed numerical integration where the governing equations are numerically discretized, lin-earized and solved, e.g. Refs. [18–22]. These computations are often demanding since sufficientresolution is required to approximate the gradients of temperature and composition. While thesemodels are valuable for understanding quasi-steady burning behavior of isolated droplets, they

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8th US Combustion Meeting – Paper # 070HE-0396 Topic: Hydrocarbon Droplet Combustion

contain far too many degrees of freedom for incorporation into a larger spray calculation for whichmillions of droplets (or groups of droplets) are transported.

The second category of models are entirely analytical. This class of model relies on layers ofapproximations to obtain a closed form solution. This category includes the model developed byLaw and Law [17] to account for variable transport properties. In their model, the transport prop-erties are first assumed separable in terms of their temperature and composition dependence. Thedependence of each function is assumed to follow a particular form (often linear). The functionalform of the mixing rules for transport properties also has to be chosen with with care to allow fora tractable result. At the end of the analysis, numerical integration is still required for the mostgeneral description.

A key finding from these previous studies is the importance of accounting for changes in thermaland transport properties to accurately predict burning rate, flame stand-off and flame temperature.The objective of the present study is to devise a hydrocarbon droplet model that allows for a gen-eral description of property variation, yet is computationally efficient so it may be used in largerscale spray simulations. The approach is based on a newly developed multi-zone, semi-analyticalapproach where local convection-diffusion solutions are used to to describe the variation of tem-perature and composition (species mass fractions) within each zone. Analytical results are derivedto determine the flame temperature and stand-off ratio in the context of a flame-sheet approxima-tion which allows for the use of an adaptive, flame-fitted zone “mesh”. The solution algorithm forthe resulting eigenvalue problem is straight forward, requiring no matrix inversion and thereforeis computationally efficient. The multi-zone description can also potentially be dynamically ad-justed so as to reduce the degrees of freedom to the limit of a two-zone model where accurate localestimates of burn rates are not required.

The remainder of this study begins with the mathematical formulation of the multi-zone model.Results are then presented with comparison to experimental data. Finally, conclusions are drawnand a summary of findings is presented.

2 Model Formulation

The particle is assumed to be spherically symmetric with an infinitely thin flame sheet located atr = rf (see Fig. 1). The single step reaction mechanism is simply mF + νOmO →

∑νpmp.

Assuming the species only diffuse from concentration gradients (Fickian diffusion), the system ofordinary differential equations describing the steady-state flow in the inner and outer regions of theflow are,

Yim− 4πr2ρDi−mdYidr

= mi (1a)∑himi − 4πr2k

dT

dr= Q (1b)

where mi is the mass flow of the ith species, m is the overall mass flow-rate and Q is the overallenergy flow-rate (to be defined). The diffusion coefficient Di−m represents the diffusion of the ith

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species in the mixture. Note, Eq. (1a) doesn’t automatically satisfy overall mass conservation, i.e.∑mi 6= m, which serves as a constraint for determining the diffusion coefficients. Various ap-

proximations exist for constraining Di−m [23, 24]. Unfortunately, none of the analytical methodsallow for each species to have its own diffusion coefficient and still satisfy overall mass conserva-tion. The simplest approximation is to assume all species have equal diffusion coefficients. Thecurrent study assumes this to be the case. The variation in transport properties due to compositionand temperature changes are accounted for by decomposing radial distribution of properties intoan arbitrary number of zones, as shown in Fig. 1. Zones 1 through f reside in the inner region, be-tween the droplet surface and the flame. Zones f+1 through∞ reside in the outer region, betweenthe flame and the far-field. The outer and inner boundaries of each zone, or shells, are defined asrk and rk−1 where k is the zone number. At the droplet surface, k = 0 ≡ s (i.e., r0 = rs). Forthe inner region, s < k ≤ f and Di−m = DF−m, where DF−m is the diffusivity of the fuel intothe rest of the mixture. For the outer region, f < k <∞ and Di−m = DO−m, where DO−m is thediffusivity of the oxidizer (O2) into the rest of the mixture.

Within each zone, the thermal and transport properties are assumed constant. The term,∑himi,

in Eq. (1b) may be expressed as: m∑εiCP,iT = mΛP T where εi ≡ mi/m is a mass flow-rate

ratio, and ΛP =∑εiCP,i is a mass flow-rate weighted specific heat. Note, ΛP is not equal to the

commonly used mass fraction weighted definition, i.e., ΛP 6= CP =∑YiCP,i. For closure of the

eigenvalue problem, εi is specified as a constraint for all species in the inner and outer regions.Equation (1a) and Eq. (1b) can be readily integrated to determine the composition and temperaturedistributions within each zone,

Yi,k−1 − εi,kYi,k − εi,k

= exp

[− m

4πrs

1

Γi,k

(1

r∗k−1

− 1

r∗k

)](2a)

ΛP,k Tk−1 − Qk/m

ΛP,k Tk − Qk/m= exp

[− m

4πrs

ΛP,k

kk

(1

r∗k−1

− 1

r∗k

)](2b)

where r∗ = r/rs is the normalized radial distance and Γi,k = ρDi−m,k. The subscripts k or k − 1indicate shell (r, Yi, T ), zone (Γi, ΛP , k), or region (εi, Q) properties. Zone k is bounded byshells k − 1 and k. From an energy balance at the fuel surface, Qin/m = ΛP,1Ts − L′vap, whereL′vap = Lvap− q′′l /m′′ is the effective latent heat of vaporization with q′′l = −kl dT/dr|l,s being theheat flux to the surface from the liquid. Neglecting heat losses from thermal radiation and finite-rate chemistry, an energy balance across the flame surface shows Qout/m = Qin/m+ εF,in∆hC =ΛP,1Ts − L′vap + εF,in∆hC , where ∆hC is the heat of combustion.

The stand-off ratio, r∗f , is determined by taking the zone ratios of Eq. (2a) for the fuel throughthe inner region and the zone ratios of Eq. (2a) for the oxidizer through the outer region with thecondition that YF,f = YO,f = 0 [25],

r∗f =

LΓO,f+1

+ 1ΓF,f

1ΓF,f

1r∗f−1

+∑f−1

k=1

(1

ΓF,k

1r∗k−1− 1

r∗k

)+ L

[1

ΓO,f+1

1r∗f+1−∑∞

f+21

ΓO,K

(1

r∗k−1− 1

r∗k

)] (3)

where the factor L ≡ ln [(εF,in − YF,s) /εF,in]/ ln [εO,out/ (εO,out − YO,∞)] depends on the concen-trations of the fuel at the droplet surface, the oxidizer in the far-field, and their relative mass flow

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rates. The flame temperature, Tf , is determined by solving both Eq. (2a) and Eq. (2b) for m in theouter region and equating them [25],

Tf =ΛP,f+1Tf+1 − Qout

m

ΛP,f+1

( εO,outεO,out − YO,∞

) 1LeO,f+1

∞∏k=f+2

(ΛP,kTk−1 − Qout

m

ΛP,kTk − Qout

m

) LeO,kLeO,k+1

+Qout

m

ΛP,f+1

(4)

The mass flow rate is determined by using the zone ratios of Eq. (2a) for the fuel and oxidizer inthe inner and outer regions, respectively,

m =

4πrsΓF ln[

εF,in

εF,s−YF,s

]4πrsΓO ln

[εO,out−YO,∞

εO,out

] (5)

where,

ΓF =

[f∑k=1

1

ΓF,k

(1

r∗k−1

− 1

r∗k

)]−1

(6a)

ΓO =

[∞∑

k=f+1

1

ΓO,k

(1

r∗k−1

− 1

r∗k

)]−1

(6b)

For fuels like n-heptane that do not absorb water into the liquid phase. The evaporating fuel is theonly species with a mass flux, mF = m. Therefore, the system is closed by defining εF,in = 1 andεi 6=F,in = 0. For alcohols, H2O dilution into the fuel is important and therefore, εH2O,in would benon-zero. In general, to account for dilution, the diffusion of water in the liquid phase is required[26–28]. In this study, the limiting case of ”perfect” dilution is considered, in which YH2O,s = 0.For equal diffusivities in the gas phase, the left side of Eq. (2a) for each species in a region is equalto a constant. Equating the products of fuel and water in the inner region and equating the productsof oxidizer and water in the outer region, both can be solved for YH2O,f . Rearranged,

εF,in =νOYF,s (YH2O,∞ − 1) + YO,∞ (νH2OYF,s − 1)

νO (YH2O,∞ − YF,s) + YO,∞ (νH2O − 1)(7)

and εH2O,in = 1 − εF,in. For the remaining species, εi 6=F ||H2O,in = 0. The outer region, for bothcases, is εO,out = −νOεF,in for the oxidizer, εp,out = εp,in + νpεF,in for the products of combustion,and εI,out = 0 for the inert species.

3 Results and Discussion

The burning of n-heptane, ethanol and methanol droplets over a range of oxygen environmentsare explored. These fuels are selected because of the wealth of experimental data available in

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the literature on burn rates (K) and flame stand-off ratio (r∗f ). Table 1 shows a comparison of themulti-zone model predictions to experimental data and estimates using a two-zone model (shown inparenthesis). A two-zone model consistently over-estimates burn rates and stand-off ratios whencompared to experimental data. At 298K, the two-zone model errors in burn rates are 56.2%,21.6% and 41.7% for n-heptane, ethanol and methanol, respectively. The multi-zone model erroris much less: 11.4%, 3.5% and 7.9%. Estimates of flame stand-off are still over-estimated, butare lower in error than the two-zone model. The equal-diffusivities assumption accounts for theunderestimation of burning rate and the over estimation of r∗f . Law and Law [16] state r∗f is a strongfunction of diffusion in the outer region. They suggest increasing the diffusivity in the outer regionwill increase burn rate, decrease stand-off ratio and increase flame temperature. Of the species inthe outer region, O2 may not be the only leading order rate-limiting species. Other estimates ofdiffusivity in the outer region could be explored.

Figure 2 shows the flame structure of n-heptane with simulation data from Manzello et al. (MCK-DDH00) [19] and Puri and Libby (PL91) [18]. Manzello et al. use a finite rate chemistry mecha-nism. Puri and Libby use a single-step reaction at a flame sheet, similar to this study, but with theaddition of a water-gas shift equilibrium reaction. Consequently, the temperature and concentra-tions of fuel and oxidizer of this study match very well with Puri and Libby. The two flame sheetstudies vary slightly from those of Manzello et al. because of finite-rate chemistry effects.

Figure 3(a) shows the flame structure of ethanol with simulation data from Norton and Dryer(ND92) [29] and Marinov (Marinov99) [30] for water being perfectly soluble in the liquid. Bothcomparison simulations use finite-rate chemistry. Figure 3(b) shows the flame structure of methanolwith water being perfectly soluble in the liquid.

Figure 4 and Figure 5 show the burn rates as a function of far-field oxygen mass fraction. The datafrom Figure 4 is from Lee et al. (LMC98) [31], Manzello et al. (MCKDDH00) [19], Okajima andKumagai (OK75) [32] and Hara and Kumagai (HK90) [33]. Although there is not much data athigher oxygen concentrations, the simulations match the existing data fairly well, especially themore recent experiments by Manzello et al. [19] and Lee et al. [31]. There is much more dataavailable for the alcohols. Figure 5 shows (a) ethanol comparisons and (b) methanol comparisons.Experimental data for ethanol comes also from references [32, 33], as well as Yozgatigil et al.(YPCKD04) [34]. Methanol experimental data comes from Marchese et al. (MDCN96) [28],Hicks et al. (HNW10) [35], Yang et al. (YJA90) [36], Lee and Law (LL92) [37], Dietrich et al.(DHDNSW96) [38] and Okai et al. (OMATKSDW00) [39]. As YO,∞ increases, K appears toincrease logarithmically. Overall, the agreement of the multi-zone model predictions to the data isvery satisfactory with errors less than 20%. In general, model predictions appear to consistentlyunder-predicted the burn rate when compared to the data. While the remaining differences couldbe attributed to physical processes not accounted for in the present model (e.g., soot and radiation[34, 40], initial dilution of fuel from water vapor [27, 28, 37], finite-rate chemistry [14], unsteadyheating [13, 15], multi-dimensional effects [41], etc.), it is difficult to discern these effects giventhe apparent wide range of reported burn rates for the same pressure, temperature and oxygenenvironment.

Figure 6 shows the effectiveness of the multizone model to account for properties based on com-position of the gas. The oxidizer far-field concentration is kept at a constant mole fraction, XO, of0.21. The increasing concentration of CO2 displaces the nitrogen in the far-field. By keeping the

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oxidizer concentration constant, the flame is not being starved by increasing theCO2 concentrationand the effects are purely because of the property variations. The current study matches gasifica-tion rate, K, very well to the experimental data of Hicks et al. (HNW10) [35]. The gasificationrate decreases slightly with the increase in CO2 concentration.

4 Conclusion

A multi-zone hydrocarbon droplet burn model is developed that accounts for variations in gas-phase thermal and transport properties. The appeal of this semi-analytical approach is a modelthat good estimates of burn rates can be determined at a fraction of the cost compared to directnumerical integration approaches, but also without incurring the errors of simplified two-zonedescriptions. Model predictions of burn rates and flame stand-off are in favorable agreement withexperimental data for the fuels and oxygen environments considered, with errors less than 20%.

Acknowledgments

Support for this work has been provided by NAVAIR through the STTR phase II program undercontract N68335-10-C-0418.

Table 1: Burning rates, stand-off ratios, flame and surface temperatures for selected fuels at se-lected states. The two-zone calculation is denoted by parentheses ( ).Fuel YO,∞ T∞(K) K(mm2s−1) r∗f Tf (K) Ts(K)

model exp.

n-Heptane0.23 298 0.567 (1.00) 0.64 [31] 11.1 (19.8) 2267 (2305) 362.2 (366.9)0.23 1200 0.680 (1.24) 1.02 [42] 9.91 (18.3) 2868 (2906) 363.9 (367.9)

Ethanol 0.23 298 0.521 (0.759) 0.54 [32, 33] 5.82 (8.27) 2133 (2080) 327.7 (329.8)Methanol 0.23 298 0.497 (0.765) 0.54 [35] 7.22 (11.0) 2181 (2179) 341.7 (344.6)

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+12$%!&'()$!%$*"+#!,-+#$!&3/!!!∞0!

%&!%.!

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Figure 1: Sketch of multi-zone droplet description.

Figure 2: n-Heptane droplet flame structure showing T , YF and YO with simulation data [18, 19]. Forthis case, ∆Tmax = 1K, T∞ = 298K, P∞ = 1ATM and YO,∞ = 0.23. Water vapor insoluble in liquid.

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(a)

(b)

Figure 3: Flame structure showing (a) ethanol T , YF and YO with simulation data [29, 30] and (b)methanol T , Yi for water vapor perfectly soluble in the liquid. For this case, ∆Tmax = 1K, T∞ =298K, P∞ = 1ATM and YO,∞ = 0.23.

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Figure 4: n-heptane burn rate verses YO,∞ [19, 31–33]. For all cases ∆Tmax = 1K, T∞ = 298K andP∞ = 1ATM .

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(a)

(b)

Figure 5: (a) Ethanol [32–34] and (b) methanol [28, 35–39] burn rate verses YO,∞. For all cases∆Tmax = 1K, T∞ = 298K and P∞ = 1ATM .

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Figure 6: Methanol burn rate verses XCO2,∞ [35]. For all cases, ∆Tmax = 1K, T∞ = 298K, P∞ =1ATM and XO,∞ = 0.21.

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A Semi-Analytical Variable Property Droplet Combustion Model · dr = _m i (1a) X h im_ i 24ˇrk dT...

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